Newbie silly question time

[DelicateDave]

Hi, just joined to pick some brains having purchased our "forever" cottage now in our 60s. My ready to have some fun and cut out ongoing outgoing to a minimum as we enter retirement6. I have loads of ideas but I'm starting off with small projects that might or might not work, this is so I can understand what I'm doing if I move of to bigger stuff :-)

So here's the first idea & the first question;

Our hall is right in the middle of the house, the is an entry hall with stairs then a small passage to a 6ft x 7ft hallway where the are doors to the bathroom, living-room and kitchen which is a newer extension on the side. This hall, although small and no real outside walls is the coldest stop in the property. My idea is to put low voltage lights in a new ceiling & insert a small ventilation fan in the living room wall. The idea being the fan will be high up near the ceiling sucking the warm air from the living room into the hall. Thoughts on this please.

Question - I've seen a solar ventilation fan that is ideal for this. But how would I combine this existing tiny solar item into my lighting solar project? the
tiny solar panel would be in the living room side of the wall. I understand the fan may still work if very slowly even from the room lighting. But much better would be to wire it into the hall low voltage system - with a external solar panel charging a small batt for LEDs. Would I be able to connect both to the battery? so the fan runs at a decent speed at night? told you it was silly.