1. Two valves mean TWICE as much water, regardless of the pipe size to the valves.
2. The existing drain should handle the added flow, if it IS the minimum 2". If it is 1 1/2", then it was NEVER to code and will be too small.
3. It is NEVER a "good" idea to notch joist, but if you decide to drill holes in them, there are strict specifications as to the size and WHERE it can be done.