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Seedlings are very vulnerable and need to have their moisture levels carefully monitored. Perhaps you could find someone to check in on them while you are away.

Pinching back is a technique for some seedlings.
I see you have tomato seedlings in the back of the image and you will not need to pinch these back.
I do not know what the plant is in the foreground.

If they are leggy, it is likely due to lack of light.
These articles will help you.

https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/seedlings-after-germination.htm

https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/how-to-harden-off-your-seedlings.htm

https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/leggy-seedling.htm

https://www.gardeningknowhow.com/edible/vegetables/tomato/growing-tomatoes-from-seed.htm

You can start the hardening off process now for peas, lettuces, broccoli and all cool-season crops. Wait until the soil warms to 60F before putting out warm season veggies like tomatoes, cukes, squash and peppers, among others. If you are growing pansies they can go out now but other flowers should wait until after your frost-free date. Tiny seedlings are very fragile. Waiting until your seedlings are 3-4 inches tall and have several sets of true leaves is preferable. Depending on the size of your cells, you may have to up-pot them. If you see roots poking through the cell, that's a sign that it is time to repot. https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/how-to-harden-off-your-seedlings.htm

As long as you only put enough water in to moisten the soil. If there is extra water sitting in there, it could cause a rot problem. It is a delicate balance, but it is necessary to watch them, and keep them moist as needed.

Yes, I've hardened off plants in the way you describe and it's worked for me in the past. The "cumulative" method seems to work.

It is best to wait until seedlings are several inches (2-3) high before planting them. They will have a hard time surviving harsh sun, pelting rain, bugs and bunnies. The fact that they are tiny indicates an under-developed root system for coping with outdoor conditions. But they might do fine! After all, many things are started outside from seed and manage to make it.