There is a chance of getting an egg every 15 minutes, I believe.
So, if you don't get an egg after putting the pair for 15 minutes,
you need to wait for another 15 minutes.
You can change castform's appearance by using a weather balloon.
The form your castform takes is the same as the current weather on
PH (that you can see on the top right in the userbar).
You can't stop rumble missions mid-way so yeah, you gotta wait for
your eevee to finish its mission. If you changed your mind about
evolving your eeveee, you have the option not to evolve when it
reaches the mossy evolving rock.
I'm not entirely sure what you're asking, but you can see your
eggdex entries in your pokedex. The little egg icon before the pokemon
indicates whether you have the eggdex entry for that pokemon or
not. (solid for yes, faded out for no.) You can see the total
eggdex progress for your national dex at the very bottom. However,
there's no eggdex counters for individual regions so you have to
scroll through them to check how many you're missing in each.
Screenshot with things I talked about highlighted below just in
case I wasn't clear enough: