I agree with ecnerwal rubber pads would be best. We are a diy site and here to help you.
I prefer to use a concrete backer board screwed down to the flooring to mount tile to. Never put tile directly on chip board , OSB or most underlayments. Plywood will work but the others when wet come apart as the tile can allow moisture and then it ...
I’d treat the vertical trim like vertical trim on doors and extend it to the floor. Then, butt the base trim into it.
The vertical trim needs to be slightly thicker anyway, so that it can be dadoed to fit over the edge of the tile.
This is more a matter of opinion than anything where one way is clearly best.
I'd likely mitre the end of the board, and trim so the end of the mitre was the same thickness as the tile trim, and either butt them together or leave a gap and caulk it.