What can be in an unheated greenhouse?

To answer your question directly, this is what I'd do:
Onions (Red Baron, Ailsa Craig, Welsh Bunching) - greenhouse or outside
Leeks (not many - had rubbish germination) - greenhouse or outside
Cabbages - greenhouse
Lettuce/salad mixes (have a container of salad in the greenhouse now, but that was started out there) - greenhouse
Tomatoes - indoors
Sweetpeas - no idea
Cauliflowers - greenhouse
Artichokes - greenhouse
Allium (yes, singular - I didn't do well with these!) - greenhouse or outdoors
Zinnias - no idea, but I'm guessing greenhouse
Tomatillos - indoors
Cucumber - indoors definitely
Watermelon - indoors definitely
Lemongrass - indoors definitely
Radishes - out!
Mustard - greenhouse
Kohlrabi - out
Chillies - indoors definitely
Aubergines - indoors definitely

I haven't got all of these types sowed yet, and I've not grown caulis before, and I am by no means an expert so do tell me if you disagree