Remember that we're all just customers here. We'll try to answer as many questions as we can and also indicate which things you might have to ask a real DreamHost pre-sales support person.
You can copy over anything, but there are some things that are more complicated to copy over than others. Static html is easy, as are any configuration files. You'd need to take database dumps and restores for any application data you're storing in MySQL. What kind of web applications are you running or are your sites all html?
You should be able to either wget stuff or possibly tar it up on the old server and ftp it directly to your DreamHost server.
I don't know of a direct transfer or "backup and restore" method for email. I've downloaded it via pop or imap and then re-uploaded it to a new account using IMAP.
Because of the nature of the beast its different between disk and bandwidth. Your disk limit simply increases over time. Your monthly bandwidth limit increases over time as well, but it's not like you get credit for unused bandwidth. Like cellphone minutes, unused bandwidth expires at the end of your hosting month.
As for as I know, MySQL is the only option.
You should be able to copy your stuff to DreamHost and then initiate a switchover one site at a time. You should also be able to test things out before switching the DNS, so your downtime for any particular site would be minimal.
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