My Android keeps auto stopping after current song... probably, due to screen power saving(?)

Yeah unfortunately this is a browser thing. We are actively searching for hacks/workarounds but haven't found anything yet. In fact it's even worse with the Chrome browser, music stops playing the second you lock the screen. :(

It does not do last.fm scrobbling, though. Sent them an email about it a few days ago.

Does the default Grooveshark site let you scrobble to last.fm? If so, how do you do it?

yep, just go to the profile settings, under "connect accounts" tab you can connect it with your last.fm account.

Excellent, I learned something new, thanks.

Yes.

Might be a VIP feature, I'm not sure.