Without further knowledge, I would assume, the reply to address has been set by the sender on the mail which you are replying to, but only an empty address has been specified.

A clever idea and I thought you'd solved the problem, but in fact the problem is worse than I thought! I manually filled in the "To:" field in the reply message I described in my original question and sent it off. Afterwards no matter what message I tried to reply to, the "To:" field was no longer blank, but instead of having the sender's email it had that original replied-to address! Shutting down and restarting Thunderbird restores the blank To: field, but as soon as I send out an email, any replies I make to other messages have that email's address in the To: field, not the senders!

Need I say this is very dangerous, since if I don't notice that the To: field has the wrong address, I will be sending email to the wrong person.

I was able to "fix" the problem:

(1) Ran TBird with two relevant add-ons disabled. Still had the problem. Conclusion: TBird or an irrelevant add-on is the cause.

(2) Ran TBird in Safe Mode with all add-ons disabled. Problem went away. Conclusion: Problem is with an irrelevant add-on.

(3) Ran TBird with (again) both relevant add-ons disabled. No problem. Conclusion: Did I somehow botch Test 1?

(4) Reenabled add-ons one at a time, expecting the problem to appear again when the offending add-on was enabled. Enabled them both, and problem never appeared. Conclusion: BTHOOM!

So what add-ons were disabled to get it working? and what does BTHOOM mean?

Well I seem to have fixed it in my Thunderbird, I disabled the Display Contact Photo add-on, and then it worked, and then I re-enabled it and it still works, but now the photos are displaying which were not before.