It's clear that Ulfric killed Torygg by summoning the Thu'um, but what specific shout was it? Was it one the player can learn throughout the game?

He FUS ROH DAH'd him to the floor and pierced him with his sword!
(at least he said it was so)

According to Jarl Elisif, the now widow, Ulfric only shouted Torygg to death. There was no sword involved.

Given Ulfric uses Fus Ro Dah several times when he fights and never uses that magical one-shot kill, I'd rather believe the other version. By the nine, I hate Elisif so much... She's like a sisy version of Sansa Stark.

Maybe the king broke his neck or had the skull fractured when he was pushed violently backwards by the shout.

I heard they were having a meeting or having dinner to discuss matters when Ulfric challenged Torygg to a duel to see who was the rightful High King.

Torygg accepted but the duel was won by Ulfric who pierced Torygg's heart with his sword, then judging by peoples accounts of what happened, Ulfric either used a disintegrate shout on him (fire breath maybe) or used unrelenting force and the fall totally destroyed Torygg's body.

Ulfric himself confirms that he killed Torygg with his sword.

Th'um code forbids to use the power of the voice for petty reasons. If Ulfric admitted he used a shout to win the duel he could be viewed as a 'cheater' and someone who doesn't care for honor or moral codes of conduct. Nords, like orcs, admire people who live by a code and would never follow someone who takes shortcuts to reach his goals.

There's is no 'Thu'um code' and Ulfric obviously does not follow the Way of the Voice. Thu'um meanwhile is thoroughly ingrained in Nord tradition, and you can bet it's a part of traditional Nord duels. The claim that he 'cheated' because he had thu'um and Torygg did not is ludicrous to me.

It seems highly unlikely to me someone without thu'um would've been able to become Jarl in the old days of Skyrim, let alone High King. If Ulfric intends to hearken back to those days he proved his point exactly by using his thu'um, despite Torygg not standing a chance regardless.
There's no issue of fairness, and the charge of murder is a fabrication, but that doesn't mean it's not questionable whether what Ulfric did was right or justifiable.

He learned with the Greybeards, but left them when the Great War started. Then again, there's nothing against using the Thu'um in duels, ancient nords used to practice It, It's like forbidding magic to fight.

Then again, Ulfric out of the two leaders is the only one who fights to death and asks you to kill him. Tullius hides and tries to forfeit.

sheesh what a wimp And here i was thinking he's someone who's the "honor glory and death" guy.

At least you have Erikkur, she's got balls of steel.