Thank you all. Sorry for my delay in responding. I went on vacation and thought I'd get back to this while there, but I did not.
My ceiling joists are 7 feet, 8 inches above the concrete floor. I plan on using floating walls and putting the new joists between the current ceiling joists. However, I'll still lose some inches both for the ceiling and the floor.
15 feet in width is about the widest it can possibly be and I could lose up to a foot or so once I frame the (floating) walls. By width, I mean the wall where the screen will go is at most 15 feet wide. One wall will abut the outside concrete of the basement and one wall will abut the load bearing row of pillars in the middle of the house.
16 feet in depth is approximate and can be increased. By depth, I mean the length of the wall that is perpendicular to the wall where the screen will go. The HT will abut both a utility room and a bathroom, neither of which is built yet (still designing), so I could change the depth by even a few feet. The bathroom will likely have a small shower, but no tub, so I can adjust it somewhat. The utility room I can definitely adjust, but I also have to meet code. I'm not sure whether code will affect the utility room, but I need to have an oil tank moved there, as the oil tank is currently in the space where the HT will go. There likely are some codes about how much space there has to be around the oil tank, and I'm putting the tank into a location with an oil furnace, oil-fired hot water tank, solar cell equipment, and air handler. I assume there are codes for how much space is around those devices, but I don't know what my town requires as of yet.