Well, all doubts have been solved finally. After taking advice from quiliro
and jxself, I have done some tests with GPG (everything works fine in command
line and no errors happen, unlike with the GUIs I mentioned, and I see no
internet connection happening whatsoever) and I asked around in Tor IRC.
Here is a simple view of things for people with some similar doubts:
Torbirdy changes settings in Thunderbird to force it to use the appropriate
proxy and the appropriate privacy settings. These are enforced in each boot
of Thunderbird, so even if you mess with something you can restart and
everything will be safe again. There is a "leak" of the actual time you sent
the email (the person who receives it can see, presumably the email service
provider as well) but not of the time zone (Torbirdy forces Thunderbird time
zone to UTC). So, the receiver can see what time was it that you sent the
email, they are working on a patch for it.
As for using Enigmail, it is safe. It works correctly, you can encrypt and
import keys and such, the only "issue" is that if the Engimail plugin tries
to connect to the internet key server without using proxy settings, Torbirdy
will deny it. That's what "safe close" means. In other words, there is no
chance of revealing your identity through there. You might find some problems
in using a key server, but again, Torbirdy uses a tor hidden service as key
server, to make it safer.
So, all in all, I believe this thread might be useful for other people (even
because different aspects of GPG and email were discussed here), and I hope
the explanation I provide here is also clear enough.
I thank everyone who tried to help. The thread is now ended.