Sounds like the kid is doing good so far. Just make sure he is nursing enough. You can give the mama some probiotic for the stress and B vitamins if you have any on hand, other than that, all sounds normal😉
Thank you so much️Congratulations! What a darling! I wouldn't be shocked that nothing showed up on the ultrasound. Apparently, that's fairly common. Goats are good at hiding their babies.
I would just make sure to feed mom well, though no sudden increases in feed rations, as you probably already know, and make sure to keep an eye on her worm load, as kidding stress sometimes increases the chance of worm problems. In a few weeks, you can either start coccidia prevention for the kid, or just wait and see if he needs it. There's loads of info on that here, so you can do a search to find out more.
If the watery eye persists/gets worse, check that eye for entropion. I had a kid this year with a mildly inverted lid. I was able to fix it myself, not a big deal, but it started out looking just like that, so it's something I watch for, now.