Hope some of the pros weigh in with paint colors for you as well as lighting recommendations. Warm white sounds good. What's under that carpet? If it stays, then make sure the white paint you pick is at the beige end of white. Here are some basements in an Ideabook in my profile. There are lots more in the Ideabook if you want to check it out for layouts, furniture, etc. It looks as if you have two nice-sized zones down there, so you can do a lot--game or dining area and a TV area.

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yes, and paint the trim and ceiling too. Honestly you need to show the whole space for color inspo. Maybe a greige very light with white trim