They need a bit of sun and warmth to develop hotness. Though they should have some as they turn red - usually at their hottest as they have just turned red.

Yes, in a larger pot the plant will increase in size.

Overwintering is a bit tricky, but a cool conservatory such as you have is probably the best kind of place for it. Some inexpert friends succeeded in overwintering a chilli in just such a location - probably gave it enough inattention - and it is in a ridiculously small pot.