Hercules opting to drop his original Greek name of "Heracles" makes sense on multiple levels. He understandably has no inclination to bear the name of the goddess who has tried to kill him or otherwise ruin his life from (literally) day one. Also, Hercules is an adventurer operating on modern day Earth, and as such, would want to use the most up to date version of his name, the one society at large would be most familiar with and most likely to recognize him by. Related to this, it is the name "Hercules" that is most firmly ensconced in the human lexicon, with expressions like "Herculean strength", "Herculean task" and whatnot. That would certainly appeal more to his ego.
Hercules seems to have a predilection for joining teams, like the Avengers, the Champions and such. This could be a holdover from his days of adventuring in ancient Greece, where he was not only a member of the Argonauts, but also quite often partnered with a squire or warrior like Hylas, or even the leader of his own small army. But another reason he prefers to be a part of teams is that he dislikes being alone with dark thoughts of his tragic history, or he may just want people around like the Avengers who can stop him if he's ever driven to murderous madness again, such as when Hera caused him to kill his own family.
A good chunk of Hercules' bristling under the Wasp's leadership of the Avengers can indeed be chalked up to misogyny, but something else to consider is that it had been quite a few years since the Lion of Olympus' last stint with Earth's Mightiest Heroes (a short stay during the Korvac saga, unless I'm mistaken). At that time, Janet van Dyne, while still a courageous hero, was still very much the ditzy, frivolous dilettante, constantly fretting over fashion and parties, trying to make her then-husband Hank Pym jealous by swooning over Thor and others. Hercules wouldn't know of the trials and tribulations the Wasp had gone through in the years he'd been gone (Hank's mental breakdown and disgrace, and the subsequent dissolution of their marriage) that helped make her a stronger, more focused person and a great leader. Anybody in their right mind would be rather reluctant to follow old scatterbrained Janet, and Hercules, rather stubborn and set in his way of thinking, would have a great deal of trouble accepting that this Wasp has matured and grown beyond the one he knew. Other Tropers have noted that during his tenure with the short-lived Champions, Hercules had no problem obeying the Black Widow, who has always been depicted as a a sharp, no-nonsense and focused utter professional, the equal to any man. We all know a probable underlying motive here for Herc's was his desire to bed Natasha, but Hercules, who has spent centuries in combat and all-out war knows full well the necessity of strong, competent leadership, and wouldn't follow someone whose capabilities he did not respect, regardless of how hot for her (or ''him?'') Herc might be.