Onions would be fine in root trainers and getting them off to an early start certainly pays off. I often start beetroot in modules - 2-3 seeds per mod, but they mustn't stay in them so long that they run out of nutrients or they will run to seed. You can have up to 4 growing in the one module, plant them out like that but with each group further apart than individual plants would be spaced - the beets still grow perfectly well squashed up like that. Carrots don't like to be transplanted so they are best sown and grown in a larger container for an early crop - I use empty licky buckets with a few holes drilled in the bottom, filled with good friable soil. They are easy to protect from carrot root fly in containers too. If you are contemplating parsnips then wait until your outside soil is ready as they really don't like to be transplanted.
Salad greens can be grown in licky buckets too, then replace them with dwarf french beans later if the soil is still wet, as the added height keeps them off the ground.