Remember and Forget

Remember
The difference between "I remember locking." and "I remembered to lock." comes from the difference in meaning of 'remember'.
If you remember doing something, you recall this action. You locked and now you remember that. You remember taking the key in your hand, inserting it into the lock and turning. If you remembered to do something, you fulfilled an obligation. You were suppose to do something and you did it, you did not forget.

You can say that in case of "remember doing" we do something and then we remember that, but with "remember to do", we first remember that something needs doing, and then we complete it:
------------x--------|----                    ------------|-------x-----
           doing   remember                       remember  to do

Other examples:

Please remember to pick up the kids from school today! (Don't forget!)
I did not remember to buy salt and now the dish tastes bland. (I forgot.)
Did you remember to book the tickets?

I remember going to the cinema last week but I don't remember what I watched. (I went to the cinema, I remember that.)
Do you remember ever meeting this guy? How come he knows you?
I don't remember signing such a document. (I signed this? I don't remember that!)

Forget
Likewise, "I forgot to lock." means that I did not complete a goal I had, and "Have you forgotten signing this?" is asking whether you forgot the action of signing.