There are several that will work - "which perl" returns /usr/local/bin/perl
The site is served from whatever folder you indicated in the Control Panel when you "Added the Domain". The only way "www/public_html" would show up is if you defined that as the directory for your domain.
That wouldn't be an upgrade at all, and you would actually have less control than you do now with a full shell available. I can't speak authoritatively for DreamHost, but I sincerely doubt they would ever do something like that - I hope not, because after learning my way around the DH Control Panel, I can't stand to be crippled by having to use a cPanel based system anymore.
There is a way to do that - just edit the user and change the password ... you can also change panel user passwords in your profile.
I don't know - what is the url to your site?