I'm not sure I understand the question. For example, my ISP is Verizon, but my IMAP email account is with a different supplier. Changing my ISP in no way affects my email account. So, if your IMAP account's server are provided by your ISP and you change ISPs, then I imagine you would have a different email id. If your account servers are with a different provider, then you should be okay. Regardless, if needed, you can click File>offline>work offline and TB will prompt you to ask if you want all messages downloaded. So, you can save your messages if it comes to that.

what happens when you change ISPs and get a new server? Are all of the emails sent / received from old ISP lost?

Nothing is lost as long as you do still have access to the old ISP account. You'll have to download your entire email archive from the server to your Local Folders account. Once the new ISP account is available you can upload your messages to the new server.

Since this procedure can be quite an effort and possibly time consuming it is recommended not to use the email service offered by your ISP. Use an email provider independent from your ISP instead, so that you don't run into the issue when changing ISPs in the first place.

Also, make sure not to edit the existing account. Create a new account for the new server.

Is everything from your computer copied to the new server?

Not automatically. You'll have to do that manually.