But now that I have played a few matches with different mechs and loadouts, and the lists are getting longer, I was trying to figure out how they were organized.
Well, it's simple, they are NOT.
Please look at the linked screenshots (or your own stats page for more meaningful numbers).
Lets start with Weapons.

Are they ordered by :
Name? : No
Type? : No
Matches : No
Shots fired : No
Hits, Accuracy, Time equipped or Damage? : No, No, No and NO!
Order I chose to use them first since the patch?? : Not even that.
What kind of a mess is that? How am I supposed to find the data I want?
The only good thing right now is that the order does not change, new insertions notwithstanding. So I can find things using spatial memory. Thats a rather weak way to present tabular information though
And its the same thing for Mechs.

At least variants seem grouped (might be accidental in my case, please confirm if you run several variants of some chassis).
But how about sorting by name? Or weight class? Or anything else?