You don't say where you live, but I suspect you are in the US. As Growing Veggies mentioned, your soil looks like garden soil and not a growing medium. Here's what the situation looks like to me.
Your soil appears compacted and it looks like the seeds had a difficult time breaking through. Some cotyledons seem to still have their seed covering stuck to them, or maybe they are just deformed. I was a bit surprised to know that you've had them under lights for 5 1/2 weeks and only have the cotyledons. These will eventually wither away when the first set of true leaves appear. Yours look chlorotic. I suspect from putting them outdoors and leaving them out for the night. You don't say if they were in full sun or not, but it looks that way. If the rain was a heavy rain, that could have damaged the stems and also flush any nutrients from the soil. I can't tell what the bug is from the photo, but it almost looks like a mosquito. The purple of the stems is normal in some varieties of tomatoes, but could be from the stress of the sudden cold temps at night.
If you put them back under lights, I suggest you have the lights about 4" from the tops of the seedlings. Have a fan blowing GENTLY in the room so there is air movement, but not blowing directly on the seedlings. This will help to strengthen the stems.
Some of the stems look broken, as if by wind or an animal or person. You might want to use a small scissors to cut at the soil, any that are broken. I wouldn't fret over a few chewed leaves. Once you have your first set of true leaves I would recommend you fertilize with an organic fertilizer such as fish emulsion mixed with seaweed.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.