There probably is some dust on the mirror, and it's very unlikely that it causes visible spots. The spots you see are likely dust on the focusing screen, which you should be able to remove and blow or brush off. The mirror surface is fairly delicate, and brushing it isn't a good idea, IMO. If you decide you really need to, do it with a brush you (or anyone else) have never "petted".
The surface of the screen is delicate too, but not quite so much as the mirror. In either case, do the minimum that gets you the desired result.
A baby's ear syringe works well as a small blower for things like this.
If the dust spots are minor then I would leave them there until I could be certain of cleaning it properly without messing anything else up or sending it out for a good clean.
They will not appear on you pictures because film gives you a new 'sensor' for every shot!
Not sure if these things have been mentioned:
You may want to check to make sure you're AF selector switch (just below lens release button) is not stuck between modes. You mentioned that you were using S mode. You may want to try C, just to check if it's not the switch going bad.
Custom function 4 allows you to switch between activating AF via both shutter release and AF-On buttons (4-0) or just the AF-On button (4-1). For testing purposes, I'd keep it at 4-0 and see if you can activate the AF with both buttons.
Clean the electrical contacts on the body (just inside the top of the lens mount) with an eraser. Even if they look clean, give em a rub. Sometimes those contacts get a bit of a film on them and communications between lens and body seem to break down a bit.
As others have stated, don't touch the mirror. You can have a technician clean it if you wish, but don't ever try doing it yourself.
dust spots: dismount the lens, look through the viewfinder on a lit white uniform surface (like a wall). if you still see the dust specks, use a blower on the mirror and inside of the prism (fosucing screen). generaly you can ignore these, they won't be visible on the photos.
if the dust specks are not visible without the lens, clean the lens front and back elements.