Vanity shelf interfering with trap install
You are going to have problems with that plastic drain.
Here's what I do :
Slide the rubber washer as far down as it will go without disengaging the nut.
Run a generous bead of clear caulking around the top edge of that rubber washer.
This way when you tighten it against the sink, the washer spreads the caulking through the threads.
If you don't do this, water will weep down through the threads. The plastic is not strong enough or the washer is not squishy enough, i dont know which, but they are junk.
The slip joint connections will not require teflon or pipe dope.