Well, I've noticed that a while ago, but I thought this would get away by itself...

So, some of (always the same) my TI-83+'es buttons sometimes get "stuck", when you press them you need to press a bit stronger and they make a clack. Also, very rarely, they can remain in pressed position after you pressed and released your finger. This can be fixed by moving them around a bit but it's still annoying.
My calculator wasn't new when I got it, in fact it was pretty old, and I don't know what might have caused this. Perhaps using it a lot.

Anyway, can this be fixed? Should I lube them?

Sounds like there's soda or something in the keys. Take the calculator apart all the way down to the mainboard, remove the rubber key mat, and break out some rubbing alcohol, paper towels, and Q-tips. Spill the keys out of the calculator, clean the sides of each key, clean the key mat, and thoroughly clean the holes for the keys in the front housing. I did this with my TI-84 Plus as I was having this issue with it, it took about an hour but it works as good as new.

Make sure you get the keys reinstalled in the correct order and orientation; you can leave the key mat in the calculator, and while holding it in place, flip the housing into your non-dominant hand and the keys will stay on the keymat. From here you can easily slide the key mat onto a table. If you do it this way, clean the calculator's front housing first and place each cleaned key in the corresponding (mirrored) hole. The keys are also asymmetrical, make note of which side points to the left/right. Also make sure when cleaning the keys that you get the bottoms really well, as grime hides in it. The corresponding area on the front housing (the interior edges of the key holes) builds up that grime too.

Well, I guess I will just use an emulator to put the keys back in there...

Is there anything else I could do wrong when (dis-)assembling it?

Be nice to the LCD (those ribbon cables don't go back on should they get pulled off), make sure you put the correct screws in the correct spots, and don't forget to remove the backup battery screw. The housing will separate by pushing the rear housing away from the front housing. The housing is held together by five or so plastic clips, two on each side and one on the top. The lower two are around where the groove for the slide case ends, and these are the only two that really matter. The clips are arranged like this:


Don't force the calculator open, but it may need a fair amount of force to push open the clips.

Thank you for this. But I am still stuck opening it: I removed all the screws, including the battery one, but doesn't open... I tried pulling the rear part upwards and downwards but nothing of both made it open. I am afraid I will break something if I will apply even more force (I was pulling really strongly).

The housing separates from the left and right sides. Once you get the screws out (all 7), the housing will very slightly separate at the bottom. Keep separating the housing up the sides, and it will get tighter as you get closer to the first clip holding the housing together, which is near where the slide case groove begins. I used a metal spudger to give some leverage as I pushed the clip open (push the battery side of the housing inwards, the housing will pop apart from the spudger's pressure). I recommend using a plastic spudger as it won't mark the housing up.