At what stage of growth may new seedling be transplanted to larger container?
Any stage. I regularly move germinated seed from peat or vermiculite to growing medium. them move these on to little pots, or different positions in 6 pak etc. So long as you treat it kindly, it won’t even know the difference. I do the same thing for seed of lettuce, tomatoes, chinese cabbage. So do commercial greenhouse growers if the costs of labor for transplant are less than the value of the seeds, e.g. double petunias, rare annuals with tiny seeds etc.
I germinate mine in paper towels and baggies and as soon as they germinate they get moved into seed starter trays. From there when they get about 2 sets of true leaves I move them into 16oz. drink cups and regular potting soil. They usually stay in those cups their whole first season and go into a gallon pot the next spring and then up from there by how much they grow. You do have to be really, really gentle with them when moving them at any of these stages.