Which buttons are you talking about, any in particular?
The Little Phatty, and many devices with 'rubberized' buttons use a system of exposed copper circuit board traces and something called impregnated carbon. In short, when you press the rubber-ish (silicon) button, the formed 'cup' of the button extends until the carbon (black) bit closes the contact 'switch' momentarily. The form of the cup then retracts to it's moulded state.
These are used on the Phatty panel and Taurus III panels for sure but also on the inside of virtually all modern synth keyboards and all sorts of other devices. If you dissect any keybed manufactured after the early 1980s, you'll find two of these under every key in the form of strips of 3, 4, 5, or 6 at a time. In the keyboard use case, the first set initiates contact or key press on, then 2nd makes contact just a fraction of a second afterwards and the cpu of the synth determines velocity by measuring the time between each press.
The nemesis of each scheme is either the silicon/rubber body tearing thereby making contact or retraction unreliable, or dirt/grime getting into the fixture, making contact unreliable by requiring a harder press. A third problem exists when the copper board traces or (more likely) the impregnated carbon wears. Here is an outfit that manufactures these if you want to read just a bit about it:
http://www.topboundusa.com/silicone-rubber-keypad.html
Since the Little Phatty uses two half sheets and is otherwise, seemingly sealed, it is unlikely that dust or spilled beverage (in the case of a keyboard) will spoil the works. But normal wear and oxidation can cause the copper to tarnish or the carbon to wear, especially if the board contact points are not smooth and if repeated mechanical motion of button press against the contact point is not on a direct z-axis plane (e.g. perpendicular into the board). It should last for ages, especially since this is not a new 'technology' but maybe if you bought your Phatty 2nd hand, somebody kept it on the deck of their oceanfront vacation home. Don't know.
It would be good to inspect and clean but definitely be careful NOT scrape the black off of the buttons but do clean them if you can, definitely DO (very carefully) clean the board with q-tips and alcohol solution. I've had great luck with this (see this SynthFreq video
http://www.youtube.com/watch?v=-czedYcY1mA ) on my Matrix 6 keyboard, it literally brought several keys back to life and improved the reliability of misfiring keys. DO check with Moog on their recommendations. You can buy a new panel if there is noticeable wear either from Moog or here:
http://www.instrumentalparts.com/kbd-04-000-0001.html and
http://www.instrumentalparts.com/kbd-04-000-0002.html.
Fun fact: did you know that the Taurus III pedals uses the Little Phatty kbd-04-000-002 part? (yes, the OSC and Env gen buttons are flipped upside down for use on the Taurus III)!!!
With regard to the question about the 'self-tapping' screws. This is normal though it's possible that a prior owner took yours apart and didn't put it back together properly. The side panels should be held together with 3 self tapping screws which are used to connect the side to the aluminum extrusion (back panel); and the remainder per side are machine screws.
Whatever you do, be careful, and CONSULT MOOG TECH SUPPORT before doing anything and/or proceed at your own risk. I have a long history of taking apart, messing with, and re-assembling gear and a 99% success rate. The 1% was the time I took apart a watch and couldn't get it back together... you only do that once!
FWIW: I asked which buttons are not quite working well because if it's the envelope generator buttons and you are accustomed to having these in EGR Auto mode (holding a button while turning the value pot), this is normal. Turn EGR scanning to ON if this bugs you. Not sure if this applies to the Stage edition but it is worth mentioning.