Sworn swords of the Rock
First of all, the raiders from Westerlands were Clegane troops who did not need any persuasion from Tywin about looting. They were sworn to Casterly Rock as regular levies/soldiers, they weren't Mercenaries like Lord Goat or Clansmen from the Vale. (But Clegane did promise them riches however.)
Robert I Baratheon's influence
Robert was a strong Monarch who had recently broken rebellion of Balon Greyjoy. No doubt Riverrun did believe that in event of a war, the Iron throne would intervene which was the reason behind the slow mobilization of Tully banners, this false sense of security. However Tullys did send a force of 4,000 men with Lords Vance and Piper in command at Golden tooth which is one direct Road from heart of Westerlands to Riverlands. Defending a narrow pass is much easier than a pitched battle because a few thousand determined troops defending such a terrain can nullify numerical advantage of their enemy (Even though not indefinitely unless your enemies are utterly incompetent). At that time, Tully host was at riverrun. After Clegane circumvented the Tully guards at the Tooth, Edmure being the tactless fool that he was, dispersed, not disbanded his armies to stop the raiders, no doubt ascertained that in case it were to boil into an all-out-war, Robert would make it stop.
Edmure had a point to be fair. If they could intercept the raiders and capture one of them alive, they could get a confession out of the captive that he was sent by Lannisters. Then Tullys could formally complain to Iron Throne that Lannisters had broken King's Peace.
Tywin Lannister's objectives
Tywin Lannister did not intend to fight an all-out war any ways. His objective was to capture Eddard Stark if he was fool enough to lead his men himself (Which he would have done if not for his injured leg, courtesy of Jaime). That way he could punish Catelyn, save his house's honor by ransoming Tyrion back and if Iron throne intervened, he would have found some pretext that Tullys initiated the hostilities as there was no proof that bannerless raiders were Westerlanders.
Death of Robert and Jaime's command
What Riverrun did not imagine was death of Robert and ascension of Prince Joffrey Totally-Baratheon to the iron throne. Remember, Tywin Lannister did not openly take the field while Robert drew breath, he was not a fool. Another thing Tullys did not expect was that Jaime Lannister would be the one to open the action at Golden Tooth, probably assuming that he was serving in KL as KG. Jaime was no political genius but by all accounts he was an excellent military leader albeit rash and prone to using brute-force instead of smart battle tactics. 15,000 Westerlanders cut their way through the meagre 4000 garrison at Golden Tooth.
Battle of Riverrun
The Battle of Riverrun in which Edmund was captured was not after some hasty regrouping of Tully Forces. Tullys had called their banners much earlier and the 4,000 Token guards were chosen from the Riverlander banners. Even vassals as far as Harrenhal had sent their levies to Riverrun which is implied because Shella Whent surrendered Harrenhal due to lack of defenders as her forces were in Riverrun with her liege lord. It was the entire available strength of House Tully (Minus the power of Houses Frey & Mallister and forces sent to defend the border regions) which got smashed by Jaime. Westerlands were richer than riverlands and Tywin Lannister paid generously to get his troops well-equipped so they had that advantage as well. From Chapter 31 AGOT, Tyrion observing boots of his slain man:
They were good boots, as befit one of Lord Tywin's men; heavy leather,
oiled and supple
If Tywin cared so much about his soldiers' boots, imagine what he did for their swords and equipment. (But one must keep in mind the difference between Sworn Swords and Peasant levies). Not to mention the excellent leadership of Jaime Lannister.
So Edmure did not harrass Jaime during siege at all, seeing as siege began after battle of Riverrun and Edmure was taken captive in that.
However here Jaime lost the advantage. He smashed the Tully Forces but he could not destroy them. Instead of playing it slow and annihilate the opposition with encirclement tactics, he collided head-on, as a result Tullys were defeated not destroyed. Scattered, not broken. Marq Piper and Karl Vance continued guerrilla attacks and Lords of Seagard and The Twins had entire of their strengths unscathed.
Numerical disparity among combatants
As for complete Numerical advantage, we do not know. We know Robb went South with 23,000 swords roughly. We know Lannister strength was 35,000 men divided into two hosts of 20,000 and 15,000. We however do not know Strength of house Tully. It would be a safe guess to assume that Riverlands could muster 24,000-25,000 swords at least because it was richer and more populated than North at least, if not Westerlands. In all conclusion, together Starks and Tullys would have outnumbered Lannisters. Tywin was too seasoned a commander to allow that to happen, so he seized the initiative and routed the riverlords before they had a chance to join their full strengths to Starks. Even while doing that he ensured that there was his host lying strategically between Riverlands and KL.
Stark relief force and further Tully follies
After that, Starks come into scene. Jaime's host gets destroyed. Fast Forward, Robb goes off to burn Westerlands. That's when Edmure makes the ultimate error of sending a force of 11,000 men to defend different border sectors. Even at that point he had not disbanded his armies. He simply dispersed them to fight small local action instead of waiting for a large pitched battle.
Hoster Tully's role
Hoster Tully was in no condition to think about warfare. He was drifting into dreams about Tansy and past and his mental health was devastated. Which left Riverlands into unexperienced hands of Edmure with not even his uncle to counsel him. He allowed the lords to leave for their lands and he sent more forces to fight along side them. Which essentially dispersed the Tully strength. But that was after all Edmure's plan all along, due to his lack of skill in field of strategy and warfare.
- Edmure and Hoster did not disband their forces at the beginning. They had called the banners as soon as they heard of Lannister mobilization. They were however wrong to be lulled into false sense of security by presence of Robert on throne. The force guarding the Golden tooth should not have been there or should have been ordered to fight a retreat in case of a bigger adversary with main role as scouts. Moreover they should have posted archers on high ground on mountains surrounding the golden tooth but instead they gave a battle which is called "Battle in the hills below Golden Tooth". When you try to fight a hopeless battle without any defined role and without exploiting the terrain for your advantage, results are catastrophic. Other lords should not have been dispersed either.
- Edmure did not harass Jaime during the siege of Riverrun because Siege did not begin until Jaime had scattered the main Tully host and captured Edmure.
- Hoster Tully was terminally ill and lost in his past. He was in no fit way to think up strategy.
- Edmure was no Robb or Jaime or Tywin. He was utterly incompetent when it came to warfare. Therefore he made the grave error which lead House Tully and Stark to grave. He dispersed his forces to fight local actions and throw Lannisters back. Again, he did not disband his Forces.
- It was not only incompetence. He personally wanted to prove himself a great warrior due to envy. He was seeing his Child-King Nephew to grow into a legendary warrior and he was envious of that. As he says in ASOS, Chapter XIV:
"What, is no one to win victories but the Young Wolf? Did I steal some
glory meant for you, Robb?"
He wanted to win some victories of his own. He lacked the sense to
see however that those victories would lead to his ultimate defeat.