1. Depends on the comfort of the child. I've had kids up being led around as young as a 1.5 and others that wouldn't even sit on a horse until 2.5 or 3. With younger kids or kids with bad balance I like to have an adult next to them, helping keep them balanced. My neice is almost 6 and she has started (barely) trotting occaisionally bareback on the lead.
2. Again depends on the child AND the horse. I don't currently have a horse I'd trust and my neice understands a lot she doesn't have all the necessary skills. My neice will be taking riding lessons this fall/winter/spring and both of my horses will be in heavy training to see who she'll start riding next year. I don't really like throwing kids on some bad tempered pony and saying "go ahead and have fun". It isn't really fair to the child or the pony, the kid should have an understanding of how their bodies effect the horse. I don't like the "get on and beat the pony or yank on it's mouth" attitude.
3. Right from the start. As soon as they are around horses they should be learning about how to properly lead, pick up feet, walk around them, brush, etc. Not doing it all themselves of course, but watching and helping you where appropriate. Whenever my neice came out to feed with me I would be talking to her about horse care, body parts, feeding, etc. I have a 3 yr old and my neice taking horse lessons with me every week. Both of them "help" me clean out the hooves, groom, and feed. My neice is pretty good already, her body isn't quite strong enough though. The 3 yo is a quick learner too.
4. I think they should, but I suck at it.
5. Minis aren't great for riding so unless you don't want the kids ever riding it, sure. Otherwise I would agree with the others that an older horse/pony is perfect. They can learn groundwork, care, and take "pony" rides. Just make sure the older horse doesn't have any arthritis issues that make it unsafe to be around. My old girl was extremely prone to tripping and had great difficulty picking up her feet, not the safest horse for kids to work around even though she was a complete sweetie and about as bombproof as you can get.
6. These are my basic rules: hard shoes, no screaming/loud noises/etc, no running, don't walk directly behind the horse, talk to the horses so they know where you are. The biggest one is: you MUST listen to me, if not you are not allowed in the barn.