I grew up with goats and have owned them for many years! Last 6 years I've had Nigerians, I love my Nigerian Goats! I would be cautious about putting them with horses, only because some will chase/kick them. I think if you can give them fence to fence time to get used to each other it will probably work to keep them together. I would recommend a pair of goats, so they can buddy with each other if they will be living with more than one horse. I would separate to feed grain, not because it would harm either, but because of potential of stealing each others feed. I wouldn't want my Minis eating goat grain, only because they would get too fat! And the goats shouldn't have joint supplements, there are some ingredients that are not safe for goats, and also beware of medicated goat grain that is not safe for horses.
You will need to feed some alfalfa either hay or pellets along with grain if you are milking, they will need the calcium to produce milk and stay in good condition. If you want all the milk for your own use, you will have to buy an alternative to feed the kids, or milk and bottle feed them. It can be time consuming to milk & bottle feed. And unless you purchase a bred doe, you will need to find a way to get her bred either by AI or place that will accept outside does for breeding. Can be hard to find for small scale breeder.
I've milked dairy goats a lot in the past, but haven't tried my Nigerians yet. I know a lot of people do milk them and it works really well because it's not such a large volume of milk. The Nigerians do have richer milk, so will be creamier than standard goat milk. If you want milk that doesn't taste "goaty" I would stick with Nigerian, Alpine or Nubian breeds. Toggenburgs and some Saanans will have stronger flavored milk. It's also important to cool it right away (put in freezer for 15 mins) after milking that helps keep the flavor better too.
Copper is not toxic to goats! They need copper, sheep can't have copper. Goats are more sensitive to mineral deficiencies so they should have loose minerals available free choice, and baking soda free choice which allows them to balance the ph of their rumen. And of course a salt block.
Goats also do better with hay fed in racks or nets off the ground, if fed hay on the ground like horses they will waste a lot of hay. I sometimes use big tubs, but they will still waste some. Goats are great in pasture to compliment horses because they will eat weeds/brush over grass (they will graze if there is nothing else, but not graze grass short like horses).
There is lots more I could write, let me know if you have other questions!