1. You're going to get a ton of opinions now!
For general patching and repair work, light remodeling work and small rooms (garages, bathrooms, single rooms) I like general purpose. I even like to do my finish coats with general purpose.
Lightweight in powder form and in a box, New construction, large remodeling jobs. Powdered is best for guys who haul it around everywhere, job to job. The weight savings from not having any water in it, alone, is an advantage.
There is also fast set. This comes in "time to set" formulas. This stuff sets HARD. It's best used for the first 2 coats where you need to pile it on to get the low spots built-up. Then use something easier to work with to finsh up and do the top coats.
2. Paper or mesh. My opinion
Mesh; anything deep, tapered edges, ceiling to wall corners, all corners for the extra strength. When I suspect an outside corner will give me a problem, I use mesh down the edges of metal corner beads.
Paper; butt joints, patch work where appearance is most important (near light fixtures). If the drywall joint is well supported (both sides fastened to the same stud/joist all the way down an edge), paper is fine. For joints where the drywall joint is spanning from stud to stud/joist to joist, paper is OK, but I prefer mesh for the added strength.
There are many situations that I have not covered, and many that I have not encountered. There are so many opinions, even on patch work and general drywall, that I dare not claim to be solefully and singly right. I am anxious to hear the other opinions too.
Tom in KY, where drywall work is referred to as; "%!@#ing old drywall $#*@!!!!" (edited for content, this is a family oriented forum)