Yes, it will be a fully functional install of Discourse, most importantly, with working email, since that’s what tends to trip people up. Apparently email configuration is the hardest problem in Computer Science these days.
All you need is skill with a web browser to alter site settings. If you do know basic CSS and HTML that will help with advanced theming, though.
The plan has to do with disk space and bandwidth, which is specific to the hosting provider – you can check Digital Ocean for their limits in those areas.
None of these. It’s apples and oranges as with the hosting plan you’re paying for the computer service and Digital Ocean doesn’t know or care what you do with it, and since Discourse is Open Source, that’s all yours too, and no one knows or cares how many staff members you have or what extensions you install.
You can check out Digital Ocean’s pricing here. For a small community (it’s not clear exactly what that means), you can get by with the $10/month option (1GB of ram). The $20/month plan will be a little faster, especially if you have lots of simultaneous users.