Definitely possible, but may not be super practical if its your daily, but if you're up for it, go for it.
Bk2 manual trans tend to be stronger and longer lasting if you can find it.
There is a long list of excess parts you would need, but they all can be taken off of a donor manual genesis if you have access to one.
This is the general list (other parts may be necessary)-
Clutch and Flywheel
Coiled Clutch line
Clutch Line (End of coiled line to transmission)
Bleeder valve for clutch line
Brake pedal (either get a new one, or cut your previous automatic one)
Clutch Fluid Reservoir
Shifter Linkage and Bracket
Shift Boot and Knob
A semi-well documented video on this can be found by Robbie Ferreira where he manual swaps his genesis. Seems like he is able to for-go the Auto TCM and just run the original wiring harness, but this could cause issues in your case.
As of the push button start, I have no clue. All automatics have a park sensor switch in the shifter housing which needs to be bypassed, however I an unaware if there are any other issues causes by the push to start.
Yes the bk1 and bk2 trannys are interchangeable between generations, but not between engines. Make sure you get the right transmission, clutch, flywheel, and drive shaft for the 2.0t as they are all different from the 3.8.