Do these three sentences make any sense?
Yes; although (2) and (3) are more open to interpretation, possibly make reference other folks' annoyance
Are there alternative ways
Doubtless dozens or scores
I thought, "Annoyance is his natural condition," but then backed off owing to equivocation/ambiguity. Tho "...natural state" better but not by much
"For him, it's natural to be annoyed," "For him, being annoyed is natural," "Being annoyed for him is natural," all equivocal.
"Natural it is, for him to be annoyed" Sort of kidding. But maybe you'll find something here: