There is value in having a full bath on the same floor if you plan on having a bedroom on that floor. It isn’t necessarily about having guests often, it is about resale value. I have seen a lot of home remodels, and there is nothing more annoying than someone having to climb stairs to use a shower. If you have the ability to do it now, it worth doing it.

Also keep in mind that if you chose to not put in a tub/shower, then how close is the is it to the bedroom. I have seen houses where the basement had a bedroom, with the closest actual full bath being on the second floor. The first floor only had a powder room. In that case a guest had to walk up two flights of stairs to use the shower. I would absolutely pass on a house without a close full bathroom to a bedroom, even if it is only for guests.

On another note, personally I would avoid carpet in the basement. If you are set on carpet, just make sure you have an adequate moisture barrier installed.

Technical answer is yes…. Should you, probably not.

Unless you are hiding an engineering degree that you haven’t disclosed, you should find a professional to look at it.

That said, you can remove that joist, as long as there is enough support on the neighboring joists to appropriately handle the load. This may include strengthening the neighboring joists, or moving them.