Push your dining table up by the built in seating leaving room for a small sofa and possibly one modern feature chair. At meal times, they can sit at the built in seating as it will make them higher up at the table. Define the 'lounge' area with a rug. Keep the children's area over by the shelving and have a small low desk area included in the shelving system for them to sit at which would remove the need for kids table and chairs.

I have built in seating around radiators in both my sitting room and hallway and we have triple glazing and external insulation. The seats are designed to be able to lift the covers to let heat out if needed. It’s comfy to sit in and we have not had issues with heat build up. The rads are not as efficient this way for sure but we have had no issues with the areas being cold as a result.

Do you plan to have a TV on the wall? What other furniture do you want to include? You could opt for a chaise sofa, not as imposing a full corner sofa but gives some additional seating space if needed. Or you could consider a sofa along the wall, looking out to the garden, with a chair in the corner, by the radiator. Will depend on which other furniture options you want and TV etc?

Terrace the garden (as Sarah shows) is the best way so creat another level with steps, however this would probably need a builder or landscaper to do it properly. You could enlarge your current patio too. My garden is on two levels .