Armaegis and I overlap a lot around here. lol
E7 + E9. As stated, more versatile and will power most headphones until you go 'high end'.
First question. Yes. People would hear your music clearly next to you on the bus. And you'd hear their conversations pretty well. The sound of the bus in general would also be there.
Question 2. Amplifiers amplify. That's all they do. The reason headphone amps make a big difference is pretty much any headphone will benefit from more power. More often than not the sound becomes cleaner and more defined. Even on a headphone that doesn't need one to get LOUDER (AIAIAI TMA-1 for example)...it benefits from having more power. The veil it has is lifted and it sounds very nice. For a cheap portable look to the FiiO E6 or E7. Great place to start.
You list Electronic music in your preferences and because of that I wouldn't suggest a Grado (other than not really liking them myself :D). Within your budget I'd recommend the HD 598 if you're okay with it being open. Genres you mentioned they excel at (IMO): Jazz, Classical, Sountracks, Electronic. Other than that they're a very good all rounder that works well with any genre you throw at them. I like them for rock but I won't say they excel at it because some people prefer something else. If you want a closed headphone check out the HD 25-1 II. Another very good all rounder, and it's very portable and rugged. The only problem there is I'd call the highs their weak spot (but not by any means bad).
My electronic music is very limited. If we cut that out of the list of genres, Would you suggest different headphones? My music collection is probably about 40% rock and goes down from there (Soundtracks, classical, jazz,, in that order), so something taht performs better on that end would be nice. I'm not dismissing the HD 598's, but I am looking for more suggestions and options.
Honestly...no. I'd still recommend the 598. I would recommend a Grado but I just can't bring myself to do it most of the time. I always think 'A Grado might be good here.' and then I remember the 598 and go with that because I (personally) much preferred it.
There are some die hard Grado fans out there though. I guess it's just not right for me.
tl;dr - HD 598 is still what I'd suggest.