As soon as they've sprouted, but don't stick them straight into Texas sun til they're a bit older - Oct. might be good, in light shade at first, then only at midday after about 10 days and watch out because they'll dry out fast (even if mature ones prefer a drier mix). They should only ever be in the mix to begin with, never in water, and always watered from the top. Remove the cover as soon as they're a couple of inches tall. If you want to keep up humidity around them (I might not have undertstood your set-up) by all means do so, but don't let the roots ever sit in water. Hard to say about the broken ones.... generally another top branch will take over once you tie it to what's left of the trunk, but your trees are way too small for that, so just be patient.

I am confused about your setup as well. Typically, you would just take black pine seeds, cold stratify them in your fridge for about 60 days, and then plant them outside in small pots or in trays. The only danger is to cover them with nets to keep the birds/chipmunks/lizards away. Seems like every critter on the earth likes to eat pine seeds.

If you want to grow your pines for bonsai it is best if you remove the leader root within 90 days to force the seedling to throw off strong radial roots.

i have a bunch of seeds in a small plastic pot that came with a kit. the pot is filled with soil and the seeds were already in there, apparently all ready to begin growing. i put them in a dark room for sprouting and the sprouts that sprouted came up within a week or so, but were delicate and i mishandled them and broke them. i think another is about to sprout, and once it is up, i will remove the propagating cover and set it in my bathroom window (good for humidity).