Wanted to address your other two points:
Like with your website, you can set it up so that DreamHost thinks it is your email server. Then when the nameserver switches over, mail will be delivered to your new server instead of your old server.
Since Google searches point to specific web pages rather than just domains, you do have to be careful here. What you want to do is make sure that the specific web pages that you really care about that are returning high page rankings from Google continue to be served up on your new server. For instance, if a search for your band results in Google directing you to www.tablasband.com/index.html, you should make sure that that page is served up in Joomla! otherwise people will get errors and eventually Google will stop returning that page from its searches.
The second effect is more subtle. Among the various reasons that particular pages are highly ranked by the Google search engine is that they are linked to by other highly ranked pages. As of the last tutorial I had about how Google's ranking algorithm worked, this includes your own website's pages. So not only do you want to continue serving the pages that are being returned by searches, but you should try to continue to serve up the pages that have high page ranks to preserve your high page ranks of the pages that do get returned by searches.
To solve your first problem, you can redirect any incoming page requests to the new pages. This will also mean that you're playing nicely with other people who may have linked to your pages. I don't know if this will help with your second problem. If you're really paranoid about this you should probably consult a specialist book or expert on the subject.
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