Robert's acts after his Coronation
When Robert became King, he issued two edicts:
- Grant of Storms's End and Paramount Lordship of ancestral Baratheon lands (Stormlands) to Renly Baratheon.
- Grant of Dragonstone and it's incomes and lands to Stannis Baratheon.
By these acts:
- He separated the senior Branch of House Baratheon as House Baratheon of King's Landing.
- He laid foundation of a cadet dynasty, House Baratheon of Storm's End.
- He laid foundation of another cadet dynasty, House Baratheon of Dragonstone.
So for all intents and purposes, House Baratheon of King's Landing was now a symbol of unity of the Kingdoms and had nothing to do with Stormlands at all as the Lords of Stormlands were now sworn to Renly Baratheon. That does not mean that Robert didn't have his own contingent of Stormlander sworn swords, who were sworn directly to the crown, not to Renly.
Escort in Episode 1
You are wrong when you suggest that the escort was made entirely of Lannister soldiers. The ones leading the escort were Baratheon men at arms, holding banners of the King while the Lannisters behind them were flying colors of House Lannister. Take a look below:

As you can see, Baratheon soldiers are leading the party. If it is still not easy to differentiate, see this close up, Baratheon soldier in front and Lannister man behind:

Moreover, Baratheon men with antlers on their helms formed the line on gates before the Royals entered the castle proper with King's guards.

More Baratheon soldiers spilled in with the King, pushing aside the Stark men who were standing on the left side of the gate and took positions:

You can see the full video here.
Lannister men in the escort are due to alliance between the Lion and the stag. Lord Tywin Lannister had 100+ soldiers given to Cersei presumably when she married Robert, to act as her guards and obey her every command. In Books, their captain is Vylarr, in show his name is not mentioned.
In the books the escort is described as:
The visitors poured through the castle gates in a river of gold
and silver and polished steel, three hundred strong, a pride of
bannermen and knights, of sworn swords and freeriders. Over their
heads a dozen golden banners whipped back and forth in the
northern wind, emblazoned with the crowned stag of Baratheon.
The golden color is the chosen color for livery of Baratheon men at arms.
Why is Sandor sworn shield of Joffrey?
The Hound left his ancestral keep when his brother succeeded his father, never to return. He went to Casterly Rock who accepted his service. Later he went to King's Landing, presumably on orders of Lord Tywin since Hound was sworn to him and wasn't likely to take orders from someone else.
Prince Joffrey was quite fond of Sandor Clegane. It isn't mentioned how did Hound become his sworn shield but I would speculate that it was due to insistence of the Prince himself. This is further reinforced when Joff dismisses Barristan Selmy so that Hound could take his place as King's Guard.
What did Renly mean by 100 swords?
By Hundred Swords, Renly didn't mean he was reforming the Watch. He didn't have many soldiers himself except a small guards retinue, as his power was in Storm's End, not King's landing. He meant to assemble guards retinues from his friends like Ser Loras Tyrell and give their command to Eddard, if he agrees to act against Cersei and her children. Lord Eddard however refused and Renly saw no choice but to flee.
He didn't have 100 swords however. Following is how the event happened in Season 1 Episode 7:
Eddard: Is Lord Renly joining us?
Varys: I fear Lord Renly has left the city. He rode through the old gate an hour before dawn with Ser Loras Tyrell and some 50
retainers. Last seen galloping South in some haste.
In the books, this is how the event transpired:
"Left the city?" Ned had counted on Renly's support.
"He took his leave through a postern gate an hour before dawn,
accompanied by Ser Loras Tyrell and some fifty retainers," Varys
told them. "When last seen, they were galloping south in some haste,
no doubt bound for Storm's End or Highgarden."
So much for Renly and his hundred swords.
Why was Beric Dondarrion there?
Beric Dondarrion was there for the Hand's Tourney and just happened to be in the court when Ned gave him command to kill the Mountain. If that hadn't happened, he would have returned to his seat at Black Haven in Dornish Marches.
Why are crown's forces made up of Lannisters?
That isn't true. King's Landing's defense is responsibility of the Gold cloaks aka city watch. There is a small contingent of Lannister guards however under command of the Queen.
The Baratheon men may have deserted in wake of King Robert's death and escape of Lord Renly.
In any case, it was the City Watch, aided by Lannister guards who defended the city against Baratheons when Stannis attacked, at least in the books. In the show, The role of city watch in the defense is given very little air time (If any).
The Crown's Forces are made up of:
- Gold Cloaks. Any Westerosi can join the Gold Cloaks however, most of them are locals of King's Landing.
- The levies drawn from Lords of Crownlands who are direct vassals to the King.
If Robert were to dismiss the traditional Royal force and recruit a force of Stormlanders, that would be like Targaryens dismissing the Kingslanders and getting people from Dragonstone to KL to serve them which would hurt the balance of power in different regions of Westeros. If Robert is to be seen as the King, he has to act like a King. Which means, holding on to Crownlands for his power and never showing particular favor to one Kingdom, even if it is his own origin.
The Lannister guards aren't official part of the Crown's swords. They are rather the Queen's swords.
Where was Baratheon Army when they declared for Stannis?
It wasn't Baratheon Army per se. It was rather Stormlander Army, sworn to House Baratheon of Storm's End and its lord Renly Baratheon.
The Baratheon Army was on outskirts of Storm's End then. Stannis had surprised Renly by laying siege to Storm's End instead of King's Landing. Renly rode with his Army, made of Tyrells and Baratheon sworn Lords to defeat Stannis. Stannis however had Renly killed by sorcery. The Tyrells fled as they had no reason to fight for Stannis, unlike Renly who was married to Margaery.
The Stormlords however had no choice but to declare for Stannis. Where were they to go except to the last Baratheon? That's how Stannis got the command of Stormlander aka Baratheon army.
How Much say do soldiers get?
None. Soldiers don't choose who they fight for. They fight for their own local Landed Knights, Lords and Lords Paramount. They are normally peasants who answer the call when their liege calls for them and follow him into the battle. It is the lords who choose who will they fight for.
The administrative ranking is:
- King
- Lord Paramount
- Major Lord
- Minor Lord
- Landed Knight
- Peasant
The administrative structure is like this:
1. King
1. Lord Paramount
1. Major Lord
1. Minor Lord
1. Landed Knight
1. Peasant
2. Peasant
2. Landed Knight
1. Peasant
3. Peasant
2. Minor Lord
1. Landed Knight
1. Peasant
2. Peasant
3. Landed Knight
1. Peasant
4. Peasant
2. Major Lord
1. Minor Lord
1. Landed Knight
1. Peasant
2. Peasant
2. Landed Knight
1. Peasant
3. Peasant
3. Minor Lord
1. Landed Knight
1. Peasant
2. Peasant
4. Landed Knight
1. Peasant
5. Peasant
Landed Knights can call their peasants to arms either by themselves or by command of their liege minor lords. Minor lords can also do that except that their commands come from their liege major lords. Same is true of Major lords except that their commands come from their liege Lords Paramount. Lords Paramount answer to no one except the King.
So the Baratheon soldiers didn't quiet have a choice between the three. The Baratheon men in Dragonstone followed Stannis, Baratheon men in Storm's End followed Renly.