Gamepads (joysticks) problems

Hello everyone!
I did not find a topic about the problem of gamepads (hereinafter simply D) and decided to create a new one. If there is already, admins, please move the question there. Sorry and thanks in advance!

I have the following problem. I play on PS3 in NSF shift (I just like this game and no other)). As soon as I put the game down, I was immediately disappointed, because I discovered that the analog sticks (L2 / R2) did not work for the native G from ps3 (original). Started choosing another G and bought a thrustmaster ferrari. I played, I was glad. I only disliked the fact that its design (by the way, very convenient) allows you to hit the ring only on the machine, I did not manage to get used to driving on the mechanics. Over time, I began to notice that G himself was pressing the buttons. About once an hour at the time of the game, I changed the position of the camera. I will not say that this is a big problem, but it was annoying. At the moment, the following problem is observed - on the left analog weaving (R) G confuses left and down, i.e. I press to the left, it works down and vice versa ..

Sorry for such a long text, the questions are as follows:
1. Does the native G from ps3 have analog L2 / R2 or they simply do not work in this particular game? If so, why don't they work? How to turn on?
2. What to do with Thurstmaster? There is no longer a guarantee, it is a pity to throw it away, G is very convenient. Is it possible to somehow cure? Settings or surgery? Can the roof run like this from weak batteries?