1. You should have checked for surprise before determining the encounter distance
2. At the end of a charge, weapon length, not rolled initiative, determines first strike (however, see note under #4 below).
3. When fighting the goblins, Yrag's extra attacks aren't limited to the number of opponents -- as a 9th level fighter he should have gotten his full 9 attacks each round against the goblins, regardless of how many of them he was facing (but see note under #4 below).
4. The extra attack in the 3/2 sequence comes in the odd-numbered rounds, not the even-numbered rounds, and in those rounds there's no need to check initiative (in this case -- if Yrag had companions their actions would still be determined by the initiative roll): Yrag's first attack always occurs first, the goblinoids' attacks always occur second, and Yrag's second attack always occurs last. Likewise with his 9 attacks against the goblins -- his first 4 attacks would occur first, initiative roll would determine which was first between his 5th attack and the goblins' attacks, and then his 6-9th attacks would occur last (note: whether Yrag at the end of his charge gets all 9 of his attacks or only 1 attack (and, in the former case, whether the weapon length first strike rule applies only to the 5th attack or to all 9) is an ambiguity in the rules -- make a ruling and stick to it (because the players will remember if you don't)).
5. Normally it takes a full round to "engage" an opponent in melee (unless charging, which can only be done once every 10 rounds); in this example the combatants all fall within a loophole in that they're already within 10' of each other from round 2 on (which begs the question of why the bad guys didn't dogpile Yrag right away since they were all technically in "melee range" with him the entire time). Had the hobgoblins and bugbears been more than 10' from Yrag (say the goblins had charged 12' during their surprise segment instead of closing 6') then Yrag would've had to "burn a round" in round three to close against the hobgoblins and vice versa (unless the hobgoblins chose to charge him -- he couldn't charge them because he already charged the goblins in round 1); he would also have suffered a free "breaking off engagement" attack from the goblin. Likewise with the bugbears in round 7.
Other than that, though, everything looks spot-on.
EDIT: Actually, one more thing: since both Yrag and the goblins were closing towards each other in round 1, they should have met somewhere in the middle rather than Yrag closing the entire distance and the goblins not moving at all (Yrag has both a faster move rate than the goblins and is charging, plus he won initiative, so he should have closed a proportionately greater distance than the goblins (with 24' between them, say Yrag moves 18' to the goblins' 6')). With this, the distance between Yrag and the goblins and the other goblinoids would be 12' instead of 6', thus falling outside of the 10' melee/engagement range and requiring Yrag and the hobgoblins & bugbears to spend rounds engaging each other as described in #5 above.
