First, welcome to the forum. There is nothing cheap about owning horses
1.Can a horse live out in the pasture most of the time, and just be grass fed?
Yes, they do much better on pasture but that alone may not sustain them. They need vitamins & minerals that they may not get from pasture alone. In the winter they will require quality hay to make up for the pasture they wouldn't be getting. Some horses are easier keepers, some harder keepers. They are all different and their needs can vary greatly. Mine are on pasture but also get fed a ration balancer and various supplements, depending on the horse. Some may also require hard feed.
2. Do horses need annual veterinary and farrier visits?
Yes, vet - annual teeth floating, shots, health certificate & coggins draw. Farrier - how often also depends on individual horse. Mine are on an every 5 week schedule.
3. What minerals do horse need? This is something best discussed with a vet or equine nutritionist as it can be a very tricky thing to get right.
Other basic care requirements
-De-worming program. Some owners rotate dewormers. I have fecals done every 3 months and then deworm (or not) based on the findings.
If you plan to have one on your own property, there is pasture management to consider, fencing, shelter, water source, equipment to maintain all of it.
Horses are also very prone to injury, common saying is you could put them in bubble wrap and they'd still find a way to get hurt. Having a savings for emergency vet calls is wise and almost a necessity in my opinion.
There is much more to horse ownership and costs but that's the basics I could think of off the top of my head.