I've found the Web Audio API to be kind of a pain to work with. I'd recommend checking out a library like band.js. As for your questions, 1) I wouldn't know without seeing your code, I remember long ago needing to enable it in a browser, but those days are probably over. Post a fiddle and I'll check it out.
2) If you plan on making more games, Unity is a really good engine to learn. The benefit from learning that would probably exceed any learning curve you incurred getting sound into your game.
3) If 3d directional hearing is your goal, it's going to be a challenge. Especially since there is a lack of audio tutorials online.
Sorry this probably isn't a big help, but sound is tough. Good luck.