Tomatoes need a soil temp of 65 to 75 degrees to grow the root system.

I might need a heating mat, indoors isn't even that warm.

Can they stay out at night safely if temps are only warm in the day?

I protect from frost, but it all depends on the size and sturdiness of the seedlings. Can you put the indoor seedlings under a flourescent light, that's what I use. You can [kind of] see it in this picture.

I only planted the seeds yesterday and was going to put them out when they were about the size of yours, but this would be my first time trying to start seeds before the season, and my first try with tomatoes. I can't buy a big light yet. I was thinking if I need to protect the tomatoes and peppers or make them really warm I could 'double coldframe' them or bring them in at night maybe.

Soon as those babies pop out of the soil you will need a light source. If you have a protected spot on the patio that will give them some bright shade that might work.

They won't 'get to the size of [mine]' without a strong light source. Just a shop light from the auto parts store could work if you only have a few.

You are in zone 6? It will get cold again even though today was beautiful. It is too early for a cold frame for tomatoes. You can use a desk lamp and a compact fluorescent bulb on a few seedlings, but still would need more light as they grew big and they will by the time for you to plant outside. A shop light from Lowes with cheap 40 watt bulbs will open up a new world.