Hi there! My name is***** am a computer professional with over 15 years of experience with both Mac and Windows computers, phones, tablets, and printers. It would be my pleasure to assist you with this. In order to best help you I will need to ask a few questions. Can you tell me please what version of Thunderbird you have on your XP computer and also what version you have on your Windows & computer? Once I know that I can provide specific instructions for moving your files and for adding a signature to emails sent from your Windows 7 machine.
Questions 3 and 4 are very much related. You probably have your Thunderbird setup to use pop mail right now. With pop mail, messages are probably deleted form the server after you retrieve them with Thunderbird, so all you will see in webmail is rent email. You could choose to leave the messages on the server but then you will see lots of unread emails with no indication of which you deleted, read, flagged or filed. Any folders you created and organize on one device won't be replicated on the other devices. If you set up your email account as imap rather than pop, you can keep your email synchronized on all devices.
If you'd like, I can with your permission, remote into your computer while you watch and first, on the XP machine copy all of the pertinent files, and copy them to a thumb drive (if you have one) or a cloud drive. I could then connect to the W7 computer and copy the files there and assist with setting up the signature and setting up imap mail for you. I can do this, while you watch, via the Premium Service of Secure Remote Assistance. This is completely optional. If you do not choose to take this service, but need more help, we can continue here.