1-4, I'd suggest just leaving the left menu up all the time for now as a stopgap until you can play around with some new ideas (e.g. I might then start with a smaller menu / expanding to bigger on hover, rather than just a hint of purple). The UI looks pretty cool, but it has to be usable. I think there's a search box too, as it popped up now and then. I'd try making that always visible for a bit and see what people think.
I think 5 is perhaps a latency issue, and because there's no feedback when things are happening it seems non-responsive For instance, if it's buffering a song ready for playing, a progress bar or even a spinning circle would be nice, rather than me clicking Play a few times because I'm not sure what's happening. Likewise, when I click a song to open up the "Play or Download" right menu, nothing tends to happen a lot of the time, and I'm not sure if it has ignored my clicks, or whether it's just doing it's thing and catching up (I suspect the latter). Ideally, if you can open up the right menu (the Play or Download one) as soon as I click a song, even if you then have to do loading stuff before play works it would seem more responsive. Just remember the user needs feedback to know something is happening.
Regarding download button, perhaps "grey it out" for now, and put a "coming soon" near it or something?
Anyway, these are nitpicks that are easily fixed, I think the app itself is a pretty awesome effort.