Welcome to permies Stephanie!
Yes, you've just made a few "beginner gardening" mistakes - the important thing is to learn from them and think laterally about your options.
I'm having to make some assumptions here - you're talking about flats with individual cells? How many cells/flat?
1. Yes, it's a really good idea to only plant a single type, or very compatible types of seeds in a single flat!
Options: A) give away or
sell "mixed flats" to people who are willing to try to baby them/transplant them as needed
B) Cut your flats apart and set the cells in different boxes
C) carefully transplant out the things that need to move
D) get more ideas from someone else?
2. Yes, Veggies really need about 3 inches of depth to stop them from getting root-bound/senescent if they have to stay there too long. It sounds like you need to transplant anything that is getting big and and think *very* laterally about "where to?"
Options: A) you know those wine boxes with the
cardboard dividers in them? Could you score some of them locally and transplant into them? You'll have to cut the box apart to plant them out eventually, so make sure everything in one box can go into it's spring/summer home at the same time.
B) As much as I prefer to avoid plastic - score a bunch of 2 liter pop bottles (sorry I'm speaking Canadian), chop the tops off, poke a couple of holes 3 cm up from the bottom so you have a tiny reservoir for
water and use them as instant pots. One liter bottles might do for some, but only if they're fairly slow growers.
3. Yes, I *totally* get the space issue, but again, I've got no real idea of your resources.
Options: A) Do you have the tools/skills to cut up free
pallets to make yourself space? Do you have access to fresh manure to put in the bottom of a pallet cube and the pots of plants on top to keep them warm? Or use pallets and salvage some windows off the internet to make some quickie cold frames?
B) Can you buy safe
hay or straw bales (there are certain herbicides in some hay that you do *not* want on your
land) and stack them up in a "U" shape with the opening south to again provide a warm-ish place for pots of plants or even right in the garden, but with a "heat trap" around them? You might need to put sheets over at night for extra warmth.
C) Can you score containers with lids - juice bottles etc - that you can fill with water to moderate the night time lows? If you use glass, it can't freeze or they'll break.
Think about some of these ideas and reply with information that would help us get you through the current problems, hopefully without just creating more problems in two weeks!