I think L'ssa would have made a good squiremaster for him. It's unlikely with his likelyhood to ignore the more theory based aspects of his training that he'd have been picked by the first wing. I think a Bronzerider in 3-1 would have made him quite pleased. He was on the bronze is better train at the time.
xd Dax as a name is perfectly fine, although after graduation teen-him would have stubbornly insisted on being called D'xir.
I'd like to think that given the current situation for him they'd have remained on pleasant terms and I'm totally open to her potentially having helped him out of the constant alcohol dependency if you think that's something she'd have done.
Now onto Shylla. I like the idea of her trying to get him to work harder during Weyrlinghood on the less physical parts and him coming to view her as this sort of irritating (to him anyway) mother hen-esk figure trying to get him to his fullest potential.
I'm not sure how he'd have viewed her using Command, but something tells me he probably wasn't aware enough to really acknowledge it when it actually happened. I think he'd sympathise with her, but still see why what she did was wrong. She's welcome to have tried to get him out of his slump, but in the early stages he may have shut her out since Evmeth was Naraleth's mother and the pair of them would probably make him think of Weyrlinghood too much. He'd have gradually let her back, but there may have been unkind words said in the time between.
She can be the one to suggest wherhandling to him if you like.
How's a friendly rivalry sound for those two? I quite like a continual struggle for who could spare the best, who's green was the fastest etc. I think her attempt to be cheerful would have earned her a lashing out at from Daxirren, but I think once he managed to reign himself in he would probably seek her out to apologise, if she hadn't reappeared by that point and if she had he'd probably make a point to apologise even if she wanted to pretend it didn't happen.