All my tubs and sinks in the bathrooms have an 'overflow drain' near the top which you are unable to fill beyond. Is there any reason why that wouldn't be sufficient?
It makes the tub taller than it is deep by a little bit and doesn't look as nice are the only reason I can think of. A little bit of splashed water just isn't a big deal in most wet bathroom setups. They could also just put a drain right off the side of the tub.
Yeah, if the drain in the sink is clogged. US bathroom pipe is 1.25" which will (probably) lead to a 2" and 4" pipe eventually (depending on when, where, and how your house was built ofc). The hole in the sink leads to the same initial pipe.