Water getting in at the nail holes or past the laps is very common with felt shingles, and moss makes it worse by holding water against the surface and letting it wick sideways under the overlaps.
On the leaking areas I'd lift the suspect shingles and reseal underneath with a bitumen roofing...
If the roof is still watertight, painting is optional and is more about protecting the felt from UV than improving waterproofing. The greying is just surface weathering of the mineral finish, which is normal and not a sign it's failing.
If you do paint it, make sure the felt is sound first — no...