Thunderbird uses its own profile, it doesn't use part of Firefoxs. You can use it without having Firefox installed.
Thunderbird doesn't use/support Firefox sync. You can copy just some of the profile to another profile but you need to understand what the files are used for if you do that.
If you use IMAP accounts there is no need to synch mail since its stored in remote folders on the IMAP server. IMAP is designed to let you share mail/folders between multiple instances, unlike POP. If you use POP accounts you can copy the mbox files from one profile to another but you need to know which profile/folder has the latest mail.
A mbox file is named after the mail folder, has no file extension, and contains all of the mail for that folder. "inbox." is the mbox for a inbox folder (has the mail), "inbox.msf" is a cache of the folder listing but some flags such as whether the mail was replied to, nice to copy it but will be automatically recreated if missing and "inbox.dir" is a renamed directory used to organize the mbox files into a folder hierarchy. "sent." would be the mbox file for the sent folder ....http://kb.mozillazine.org/Files_and_fol ... hunderbird
is slightly out of date but its description of files you care about should still be good.
Don't overlook the possibility of configuring remote address books using CardDAV. If you do that for both profiles that will keep the address books in synch. Take a look at the CardBook add-on. It supports cardDAV but also replace the address book window with an enhanced version. If you don't like it you could use the "Provider for CalDAV & CardDAV" add-on instead.
If both PCs are on a LAN another possibility is to share a single profile between them. You'd just need to create the file share on one PC and mount it on the other PC. Samba would be a good way to do that. If you exit Thunderbird on one PC before using it on the other PC there shouldn't be any problems due to file locks.