go over three layers deep. Is this true? If so, should I be putting
down tar paper on top of the old shingles before laying the new layer?

Also, is there any disadvantage of going with fancier asphalt shingles
than three tab? I've seen some neat stuff from home depot
(manufacturer GAF) that look really neat, but are definitely not three
tab design.

It has been done for years, not popular in my area as high winds are
common, and the physics of the longer nails isn't as good.

The Timberline type shingles are an advantage especially for an amateur
roofer; as they ares so asymmetrical to begin with that mistakes show
up less.