1) You can run bebot in a unix shell. Best use the tool screen, which allows you to disconnect processes from the opened shell. That means you can close the connection to the server.

2) Not sure that offer is ideal, as it charges per process you run. And it's not clear if mysql/php is available on the command line.

If you aren't limited in processes, try screen out, it really helps - as you always can go back and connect to the screen log again.

Well, you could look into lomag.net

I deal with them for our irc server, and service is great. Send off an email and ask what they can do in regards to what you need.