Good that the heat/air system is over the door and doesn't interfere with your decor too much. It appears the ceiling fixture to the right of the FP might indicate that the dining space is there. Put a sofa in front of the FP and two small accent chairs with it, facing each other over a round glass coffee table. Can you wall mount the TV left of the FP? I think the room is too narrow to put the TV above the FP - you'll strain your neck. If you mount it opposite the FP, a sofa and two chairs could work but will need to be small scale - try apartment-sized seating. The sofa could face the window with the chairs opposite it. You could turn to see the fire or to watch TV. Swivel chairs come in handy in such tight space.

Need photos, but in general, I would recommend 2 chairs opposite the longer end of the sectional.

Well, for lack of a better idea, since you find it a drafty little space. I'm assuming the draft comes in on the right? Most of the necessities in the area are located of the left side; i.e. vent, thermostat, light switch etc. What I would do I think is caulk up the drafty spots, maybe even add sized poster board wrapped in batting I covered in a gorgeous fuscia with diagonal grosgrain ribbons every which way covered with a heavy velvet to be attached to the area above & below the window. (Of course remove the drapes that are hanging in there). This for the purpose of using it as a pin-it board for notes. Attach a table height board covered in felt to carry across the full width of the little room. Leaving just enough space for a comfortable little antique desk chair. Paint the back wall a bright cobalt blue, paint the ceiling a silvery pearlescent white. Find a way to attach a clip-on task lamp. And stretch a heavy duty enough wire across the top of the opening and complete it with a heavy doubled velvet drape with a drooping valance. Place a swag with fringe & jewels on it to be closed when not in use. Voila intrigue is yours!

Hey! Great space! The table is too small for the space, it would immediately help if you had head chairs on either end of the table. It would read larger. If you're interested in a new dining set, try a round dining table, seating at least 6 for size. You need something to define each space, a new furniture arrangement could help. I would try rotating the sectional so that its off of the window. Good Luck!