What is the best process for a first-timer doing a small drywall finishing job?
Paper tape or mesh tape?


It comes down to preference. As a "beginner" (but not for long... haha) I feel that mesh tape will be easier for you. No, you do not have to use any special mud; premix will work just fine. Some say that paper tape is stronger; some say that it is less likely to form hairline joint cracks over time.


  Pre-mixed vs setting-type compound.


You should absolutely use pre-mix joint compound to bed and cover your tape, and pre-mix topping compound for subsequent applications. Setting-type compound takes some special skill gained with experience, must be troweled/worked rapidly, and is harder to sand. You have time, take advantage of that to make this difficult work easier for you.


  Dust control compound?


Yes, it applies just like regular pre-mix but is less dusty when sanding. If money is an issue and dust not so much, don't bother with it.


  dust


The vacuum dust collecting sanding pads do work, but you are talking about significant expense, the vacuum is loud, and they only work with drywall sanding mesh/screens which I do not recommend for anything except the initial mud knock-down.

Tips:


Buy quality knives/trowels and set aside plenty of time to clean and dry them after each use.
Buy 3" (for scooping mud from bucket), 6", 10", and 12" knives.
Cover unused mud in buckets with plastic cling wrap (Saran Wrap) pressed tight to the mud, then snap the lid back on the bucket.
Get a strong worklight and shine it from the side, oblique to the wall, when finishing and sanding. This will highlight uneven areas and imperfections with shadows.