I myself generally don't with my flower seedlings, though others may have a better idea. Other types of seedlings may be more robust or may actually need the boost potting soil can provide.

Potting mix, of your choice, is recommended. Peat pots are highly undesirable. They suck, plain and simple...google or search for more on them.

SCG

Mix is the operative word, potting/growing mix, not soil. Soil-less potting mixes are used in containers. Soil compacts and drains poorly.

Dave

I agree I would never plant anything in peat pots and I definitely wouldn't plant pot and all.. When peat gets dry it is hard to rewet in the ground. It will simply run the water off of it. On year when I was a newbie to gardening I planted my tomatoes in them. They didn't do well and when we pulled the plants we discovered why. All the roots were contained in those pots. They never grew through the pots because the pot was never wet enough to make it penetrable.

repotting depends on the size of your cell packs and the size of the roots on your plants and how big the plant is. I plant in flats and transfer most of my plants into 6 celled cell packs when they are big enough. Tomatoes and peppers go into 16 or 18 oz solo cups. However I don't plant extra early so I don't have huge plants. I use the weeks written on the seed package as my guide for when to plant my seeds and that works for me

What would you recommend for seedling containers, for plants which dislike their roots to be disturbed?

Pot of your choice...just gently pop the plant out of it, root mass and soil, don't disturb roots...all is well.