Welcome.
-Is it intentional that my ships/colonies and shipyards MUST be doing something before I can end my turn? What if I don't want to be building anything? I have not figured out a way to end my turn without having all my things doing...things.
It seems to be intentional. Things like planets, ships, and research could be doing stuff. The alternative is you could be wasting what they could be used for. Not efficient, so game was designed to help you in that regard.
Unfortunately, sometimes it can be a pain to deal with the help. While you often want ships moving, attacking, exploring, surveying, sentry/guarding, etc, sometimes waiting is the best strategy. Unfortunately, waiting hasn't got much support. The 3 options you got is sentry, guard, and pass turn. Sentry and gaurd don't always work, and pass turn needs to be clicked every turn. Sometimes having ships do nothing is what you want them to be doing. Have them ready to do something, be in position to be able to do something, and be ready when the time comes to do something with them.
I have asked every so often for a fortify command to get around it. I've been asking since GalCiv 2. Back then, I found that using sentry or guard never worked as well for me as simply ignore the ships. Unfortunately, you can't ignore ships anymore.
Likewise, assigning projects to planets can be a work around for having to give them new orders. Unfortunately, some projects are not worth while to keep doing forever. Population growth projects aren't worth while if your population is at max.
This is still beta, so this stuff could change.
-I have spent my first 20 turns sending my ships out to explore, and I've found I think six separate stars, each with some planets...and they are all Class zeroes. All of them save the tiny one near my home planet. Is that typical?
Sometimes bad luck happens. In GalCiv 2, the game was supposed to make "balanced" maps. "Balanced" in the sense that 1 starting zone might have many planets, while another might have galactic resources, while another might be easy to defend. "Balanced" in the sense that there might exist someone who would know what to do with your starting zone and would want it. The game is still in beta, so don't expect this feature to be finished yet.
-There are asteroids that I feel like I should be able to mine for minerals/ore, but I don't have anything with which to do so. Does that come with research?
Maybe you could get stuff to use them later. Maybe they will forever exist as something that will serve as decoration or as an obstacle. I don't know what will happen with them.