Usually. Going through the overflow drain gets you straight into the drain without having to make the sharp bend in the tub drain. There could be a trap that comes apart in the drain underneath if it...
Many tub/shower valves are designed to be used either as a tub only or as a tub-shower combo. You will note that the valves have inlets on the sides for the hot & cold water supply, and outlets...
You can buy these tubs commercially.
You will have spent more money trying to build your own than you would in buying one. You also need to consider that the manufacturer might also provide warranty...