I have grown Hosta from seed. I have started mine in small regular pots of sterile soil, placed in a Ziploc bag. It is easier to start them under fluorescent lights (rather than sunlight/window). Keep them within a few inches of the light bulb. The lights don't give off a lot of heat....but if you put the closed bags (or other similar terrarium structure) in the sun they will quickly cook inside. After the plants germinate and the first leaf fills in you can slowly open the bag more and more to acclimate them to more normal conditions. If inside you may want to keep them in the bag (but with the top partially open) for some time since inside air is generally too dry. BTW you may wish to start them earlier........they start out so small and slowly that having them inside for a couple more weeks will help get them to a decent size before going dormant next fall. Also note that unless the seeds came from a streaked mother plant it is most likely you will not have variegated seedlings (though you still may get a mixture of leaf sizes/shapes and colors depending on the parent plants). Have fun with them!

I do know folks who have done hosta from seed via winter sowing outside. Been awhile but IIRC it worked quite well. They used plastic milk jugs. Doing so outside did require that they take off the cap once temps warmer up/days got sunnier to avoid cooking the seedlings.

Do hosta seeds require cold stratification? I know many plants do. I have zero idea whether that applies to hosta.

A terr should be quite doable, although, as WN mentioned, you might be better off having them under lights rather than relying on sunlight streaming through a window once they germinate. (TtBoMK, hosta seed do not require exposure to UV/sunlight for germination.). Personally, don't believe I would do a "sealed" terr. As long as the media is kept moist, there should be sufficient humidity down at seedling level. An open terr would present less risk of damping off and a less traumatic adjustment to normal humidity levels.