I assume for heating TH, you have baseboard hot water heat. You need to understand your capacity both for delivering hot water space heat and hot water, ie is hot water heater sized correctly?..

1.no, if gas lines installed correctly.

2. You need more BTUs and/or improve insulation properties to the bedroom in the basement. Any other living space in basement besides the bedroom?
How to accomplish is another issue both in practicality and economical cost.
Basements at or below grade with reasonable insulation properties usually have small heating and cooling load.

Your fuel source is nat gas?

IMO

By "compete" tigerdunes is correct with his answer, but you may be losing a lot of heat through the chimney of the fireplace. There is a lot of variability in how gas fireplaces work. If the fireplace has an open draft, you are sucking warm air out of the house to run it. The best way to supply combustion air is from a point closest to the fireplace.

Warm air from the basement is being drawn right up the stairwell. You have quite a stack effect with three floors. Cold air to replace it can either come from leaks if the hot air exits though leaks higher up. It can also cascade down the stirs. Sealing air leaks at all levels might help with that.