I'm pretty sure (almost positive) that the ACP backs up the full database. Seems to me that there are a few options, but you can select a full database backup.
Now, the database doesn't store all the information from your forum - if you were to move to a different host than DH, you'd need more than the mysql backup to restore your forum. You'll also want to use FTP to download the entire forum directory. That way you get smilies, MOD's, etc.
As for the seconed part of your question, there's no way to backup your databae through the panel. You'll have to either log into phpmyadmin and backup from there (search the forum for more specific instructions, I know this came up a few weeks ago) OR you can log into telnet/shh (like a ms-dos box) and backup the database, then use ftp to download the backup. Search the K-base for more info on how to do that.