Yeah i wouldn't start the tomatoes yet.
We grew them in the raised beds which consisted of (bottom to top) rotting wood, leaf compost, soil, straw.
Carrots aren't bothered by frost and actually grow sweeter in cool weather.
It was a test to see how they would turn out. They were slow to grow but came out pretty sweet as far as carrots go.
I've heard of people growing them under houses and only digging up what and when they needed some.