I will begin this Self-answer with a statement from GRRM, which is quoted from So Spake Martin, entry for April 29, 1999:

It was not so much a question of some king changing the rules, as you venture, as it was of the rules themselves being very variable. Medieval tourneys were never governed by a single set of rules or rulesmakers, like NCAA football or major league baseball or even (shudder) boxing. In essence, every tourney had its own rules. The lord or king who was staging the event would usually choose the format of the tournament in the broadest sense, and then appoint a "master of the games" to run the event and make all the "fine print" decisions.

In the same statement, Martin said this about Tourney of Kingslanding by Robert Baratheon:

Of the ones so far... well, the Hand's tourney at King's Landing was put together hastily, on Robert's whim, and so was relatively small, which allowed the single-elimination tilting format, which your opponents are chosen simply by the luck of the draw, and only one champion remains at the end. I also used the free-for-all last-man-standing style of melee, which did not exist in the real world so far as I know (melees were mock battles fought by teams), but which I thought offered juicy possibilities for a fantasy book.

Now to answer the question:

1. If there are no restrictions set by Game masters of the tourney, Non-knights can joust in the Tourney. Jory Cassel was not a knight but he jousted in Hand's Tourney in Kingslanding. He defeated Horas Redwyne and of one the Freys before losing his seat to lance of Lothor Brune, who wasn't a knight either. (Lothor was later knighted for valor in Battle of Blackwater bay). So it is clear that Non-Knights can sign up for jousting if the rules do not explicitly forbid them to.
2. Melees can also vary in rules. Melees can be fought in "Last-man-standing" format or between teams of Knights fighting for victory. But there is precedent of a person who wasn't a knight but still fought in melee at a tourney. That character was Prince Maegor Targaryen (Later King Maegor the Cruel) who at age of 13 fought in a Melee and defeated adult Knights. Maegor was knighted three years later at age of sixteen by his father King Aegon the Conqueror, so it is evident that when he fought in the Melee he was not a Knight. In the World of Ice and Fire, there is mention of a Melee at Last Hearth in North. Since Northmen usually do not have Knights, it is likely that most of the participants were not knights. Sadly I could not find anything on Knighthood status of Thoros of Myr.

I will end this answer with another Quote from GRRM:

As to your questions regarding the participation or non-participation of sellswords, squires, freeriders and the like, again, I don't see that as the difference as being chronological so much as geographic. The Reach is the heart of the chivalric tradition in the Seven Kingdoms, the place where knighthood is most universally esteemed, and therefore the place where the master of the games is most likely to devise and apply stringent rules. In Dorne and Storm's End and the riverlands and the Vale, things are perhaps a little less strict, and north of the Neck where the old gods still reign and knights are rare, they make up their own rules as they go along.

After waiting for almost a month, I will begin this Self-answer with a statement from GRRM, which is quoted from So Spake Martin, entry for April 29, 1999:

It was not so much a question of some king changing the rules, as you venture, as it was of the rules themselves being very variable. Medieval tourneys were never governed by a single set of rules or rulesmakers, like NCAA football or major league baseball or even (shudder) boxing. In essence, every tourney had its own rules. The lord or king who was staging the event would usually choose the format of the tournament in the broadest sense, and then appoint a "master of the games" to run the event and make all the "fine print" decisions.

In the same statement, Martin said this about Tourney of Kingslanding by Robert Baratheon:

Of the ones so far... well, the Hand's tourney at King's Landing was put together hastily, on Robert's whim, and so was relatively small, which allowed the single-elimination tilting format, which your opponents are chosen simply by the luck of the draw, and only one champion remains at the end. I also used the free-for-all last-man-standing style of melee, which did not exist in the real world so far as I know (melees were mock battles fought by teams), but which I thought offered juicy possibilities for a fantasy book.

Now to answer the question:

1. If there are no restrictions set by Game masters of the tourney, Non-knights can joust in the Tourney. Jory Cassel was not a knight but he jousted in Hand's Tourney in Kingslanding. He defeated Horas Redwyne and of one the Freys before losing his seat to lance of Lothor Brune, who wasn't a knight either. (Lothor was later knighted for valor in Battle of Blackwater bay). So it is clear that Non-Knights can sign up for jousting if the rules do not explicitly forbid them to.
2. Melees can also vary in rules. Melees can be fought in "Last-man-standing" format or between teams of Knights fighting for victory. But there is precedent of a person who wasn't a knight but still fought in melee at a tourney. That character was Prince Maegor Targaryen (Later King Maegor the Cruel) who at age of 13 fought in a Melee and defeated adult Knights. Maegor was knighted three years later at age of sixteen by his father King Aegon the Conqueror, so it is evident that when he fought in the Melee he was not a Knight. In the World of Ice and Fire, there is mention of a Melee at Last Hearth in North. Since Northmen usually do not have Knights, it is likely that most of the participants were not knights. Sadly I could not find anything on Knighthood status of Thoros of Myr.

I will end this answer with another Quote from GRRM:

As to your questions regarding the participation or non-participation of sellswords, squires, freeriders and the like, again, I don't see that as the difference as being chronological so much as geographic. The Reach is the heart of the chivalric tradition in the Seven Kingdoms, the place where knighthood is most universally esteemed, and therefore the place where the master of the games is most likely to devise and apply stringent rules. In Dorne and Storm's End and the riverlands and the Vale, things are perhaps a little less strict, and north of the Neck where the old gods still reign and knights are rare, they make up their own rules as they go along.

I will begin this Self-answer with a statement from GRRM, which is quoted from So Spake Martin, entry for April 29, 1999:

It was not so much a question of some king changing the rules, as you venture, as it was of the rules themselves being very variable. Medieval tourneys were never governed by a single set of rules or rulesmakers, like NCAA football or major league baseball or even (shudder) boxing. In essence, every tourney had its own rules. The lord or king who was staging the event would usually choose the format of the tournament in the broadest sense, and then appoint a "master of the games" to run the event and make all the "fine print" decisions.

In the same statement, Martin said this about Tourney of Kingslanding by Robert Baratheon:

Of the ones so far... well, the Hand's tourney at King's Landing was put together hastily, on Robert's whim, and so was relatively small, which allowed the single-elimination tilting format, which your opponents are chosen simply by the luck of the draw, and only one champion remains at the end. I also used the free-for-all last-man-standing style of melee, which did not exist in the real world so far as I know (melees were mock battles fought by teams), but which I thought offered juicy possibilities for a fantasy book.

Now to answer the question:

1. If there are no restrictions set by Game masters of the tourney, Non-knights can joust in the Tourney. Jory Cassel was not a knight but he jousted in Hand's Tourney in Kingslanding. He defeated Horas Redwyne and of one the Freys before losing his seat to lance of Lothor Brune, who wasn't a knight either. (Lothor was later knighted for valor in Battle of Blackwater bay). So it is clear that Non-Knights can sign up for jousting if the rules do not explicitly forbid them to.
2. Melees can also vary in rules. Melees can be fought in "Last-man-standing" format or between teams of Knights fighting for victory. But there is precedent of a person who wasn't a knight but still fought in melee at a tourney. That character was Prince Maegor Targaryen (Later King Maegor the Cruel) who at age of 13 fought in a Melee and defeated adult Knights. Maegor was knighted three years later at age of sixteen by his father King Aegon the Conqueror, so it is evident that when he fought in the Melee he was not a Knight. Sadly I could not find anything on Knighthood status of Thoros of Myr.

I will end this answer with another Quote from GRRM:

As to your questions regarding the participation or non-participation of sellswords, squires, freeriders and the like, again, I don't see that as the difference as being chronological so much as geographic. The Reach is the heart of the chivalric tradition in the Seven Kingdoms, the place where knighthood is most universally esteemed, and therefore the place where the master of the games is most likely to devise and apply stringent rules. In Dorne and Storm's End and the riverlands and the Vale, things are perhaps a little less strict, and north of the Neck where the old gods still reign and knights are rare, they make up their own rules as they go along.

I will begin this Self-answer with a statement from GRRM, which is quoted from So Spake Martin, entry for April 29, 1999:

It was not so much a question of some king changing the rules, as you venture, as it was of the rules themselves being very variable. Medieval tourneys were never governed by a single set of rules or rulesmakers, like NCAA football or major league baseball or even (shudder) boxing. In essence, every tourney had its own rules. The lord or king who was staging the event would usually choose the format of the tournament in the broadest sense, and then appoint a "master of the games" to run the event and make all the "fine print" decisions.

In the same statement, Martin said this about Tourney of Kingslanding by Robert Baratheon:

Of the ones so far... well, the Hand's tourney at King's Landing was put together hastily, on Robert's whim, and so was relatively small, which allowed the single-elimination tilting format, which your opponents are chosen simply by the luck of the draw, and only one champion remains at the end. I also used the free-for-all last-man-standing style of melee, which did not exist in the real world so far as I know (melees were mock battles fought by teams), but which I thought offered juicy possibilities for a fantasy book.

Now to answer the question:

1. If there are no restrictions set by Game masters of the tourney, Non-knights can joust in the Tourney. Jory Cassel was not a knight but he jousted in Hand's Tourney in Kingslanding. He defeated Horas Redwyne and of one the Freys before losing his seat to lance of Lothor Brune, who wasn't a knight either. (Lothor was later knighted for valor in Battle of Blackwater bay). So it is clear that Non-Knights can sign up for jousting if the rules do not explicitly forbid them to.
2. Melees can also vary in rules. Melees can be fought in "Last-man-standing" format or between teams of Knights fighting for victory. But there is precedent of a person who wasn't a knight but still fought in melee at a tourney. That character was Prince Maegor Targaryen (Later King Maegor the Cruel) who at age of 13 fought in a Melee and defeated adult Knights. Maegor was knighted three years later at age of sixteen by his father King Aegon the Conqueror, so it is evident that when he fought in the Melee he was not a Knight. In the World of Ice and Fire, there is mention of a Melee at Last Hearth in North. Since Northmen usually do not have Knights, it is likely that most of the participants were not knights. Sadly I could not find anything on Knighthood status of Thoros of Myr.

I will end this answer with another Quote from GRRM:

As to your questions regarding the participation or non-participation of sellswords, squires, freeriders and the like, again, I don't see that as the difference as being chronological so much as geographic. The Reach is the heart of the chivalric tradition in the Seven Kingdoms, the place where knighthood is most universally esteemed, and therefore the place where the master of the games is most likely to devise and apply stringent rules. In Dorne and Storm's End and the riverlands and the Vale, things are perhaps a little less strict, and north of the Neck where the old gods still reign and knights are rare, they make up their own rules as they go along.