You need to invest in a VCDS. It will be the best 200.00 you spend. You can start with error codes. The DMF will cause vibration to a certain extent, depending how bad it is. Dont throw parts and money until you diagnose, other than the DMF, its a consumable wear item on this cars. There should be codes from what I understand on the DSG's when the mechatroics start to crap out. Those codes, usually point to solenoids. I would not buy a used one, unless they have a solid guarantee that provides cash back and not yard credit, unless they have a lot of these sitting.
You need to pull the codes to start with the basics. Other issues on these cars are the chaffing of the wiring harness and loose grounds. Plenty of threads on the forum to search for. Cannot state enough, spend the 200.00 on the VCDS. You also need to factor in the Timing Belt- when was that done? Pull the VC and check the cam lobes. Tighten cam cover, assume that the whole back of your motor is covered in oil ;-) everyone of the used BRMs I have had all had leaky covers ! This is not in context to your complaint, but a preventive measure. No receipt with defined labor, assume you need to do a belt , waterpump and Tstat.
If you are pulling the DMF, replace the thermoplastic coolant part on the side of the head. You have easy access and its a great investment and solid time to do it. As others have mentioned in other posts, look at the heater hoses for softening. Check the play on the turbo while the trans it out. Check the harness - unwrap and inspect each wire down by the starter. Clean your grounds (behind headlights, under airbox, next to battery and under cowl) Those are all good preventive measures that will help you have a solid car, IMHO.
These cars can suck up some time and money initially, but when all of the common wear crap is resolved, you have a solid car that is fairly cheap to maintain. Be thankful you are mechanically inclined. Go get your VCDS and start pulling codes.