The only real problem with rain is that it could wash the soil out of the starter pots. Light rain shouldn't hurt them.

At this point (June) I would get them in the ground. I started seeds 3 weeks ago and put them in the garden a couple of days ago. They look fine in spite of a belly-washer rainstorm and very cool weather.

My mother used to have MGs every year because we kids would break the seed pods all over the ground in autumn. I'm not convinced that starting them indoors is worth the extra effort.

I started some via wintersown method earlier this year. Put them in a large planter ~1 month ago - they had 1 set of leaves (initial) on them. They're growing like crazy!