When you copy the full contents of your old mail directory to the other computer the next step is that you start TB and create new accounts with the same names as they had on the crashed system.
TB will recognize that the account exists (because the name of the account is the same as the matching account directory) and will use the existing account settings, folders, filters, etc.
You might give that a try before deleting stuff. What may have happened (I don't know what you did exactly) is that you replaced the contents of your alternate PC with the contents of your old mail directory. On starting TB will start looking for the accounts that were supposed to be there (the number of accounts and their names are stored in the registry), as they weren't there, TB decided to create them again, but kind of empty with only the system folders (Inbox, Outbox, Sent messages, Trash, Inbox Known) and zero messages of course.
So creating an account in TB with the same name as on your crashed computer should do the trick.

About the weird folders in your mail directory, I don't know what they are, but what's supposed to be included in your mail directory is this:
A directory for every account you've got, account name and directory name are the same.
In case you've been using common folders, you'll see a directory for every common folder (with identical names)
For the rest my mail directory contains directories with the names Speller, Dictionairies (both for the spell checker), hints and imgfiles
Every account directory includes directories matching the folder names for that account, unless it's an imap account, in that case there's an imap directory in the account directory and that includes all imap related caches for that account.
You mentioned that both computers had the same TB version, that's important, because there might be some differences between versions.

This whole story is valid only when you didn't use an encrypted message base. In case you used TB's OTFE (On The Fly Encryption, only available with a pro license) the names get a bit more complicated, but even then it's supposed to work if you used the same master password on both systems.