Building code height requirement is 7ft, but renovation code has a clause about working with whatever is existing. It's all about preference and you. 6'5"is not low if you are all under 6ft. Digging down requires underpinning the foundation and that is expensive. Consider painting the floor or using vinyl plank to avoid build up and keep the ceiling open and spray paint white, the space between the joists will help it to feel taller. And if you don't finish the ceiling, it will not be considered finished space for tax purposes. You can also get flush lights mounted between the joists. And yes, you may need to get doors specially made, so I would leave it as open as possible if you can. For a bathroom, you most likely will need to get a saniflo pump for a toilet, not crazy expensive but can be noisy and sometimes have issues. You will not have the height for a shower though, without trenching into the floor.