what size pots would you say suitable for each variety
Difficult to say not knowing the size of your bulbs and tubers,
You might get away with 4" pots for your lilies but I think you will need 5"/6" pots for the others.
Because you are basically storing them and starting them off I would say as a rule of thumb you should aim for at least one inch of compost all around the bulb/tuber.
So say a bulb is 2" diameter you would want a minimum of a 4" diameter pot. (1"+2"+1")
Dahlia tubers can be any shape and size and my guess is the ones you have will be quite small, so you might get away with a 4"-6" diameter pot.
When I was a member of the Dahlia Society we would refer to them as 'Chicken Legs' simply because this is what a split tuber usually ended up as.
Because of their shape in this case you would need a deep pot as opposed to a wide pot, so yes stick to the 1" of compost all around!
I hope this helps!
By the way do not subject them to very high temperatures or else they will cook and rot to mush! 8-10°C (40-50 °F) would be ample.
A steady lower temperature is much better than short bursts of high then low temperatures like you get when the heating is on and off.
