I personally don't leave halters on while horses are turned out. I have seen too many bad things. Some people do though. IF you are planning on leaving a halter on a horse while its turned out I would recommend getting a nylon halter with a leather poll strap, then if the horse got tangled in something the leather will break.
Well to answer question 1 it depends on if the horse is hard to catch or where they are turned out. If it is in a small padock an the horses can't get hung an injured themselves some people will leave them on not to mess with them. Some people also leave halters on horses that are a pain to catch cause it makes it easier. If your horses is easy to catch an in a larger pasture I would not leave a halter on it unless it was a break away halter.
To answer question 2 it depends on what you want the halter for. There is no best type of halter in my eyes. I use rope halters because I like the style and they are easy. Nylon ones can be good if you want something a bit more pro looking without blowing your pocket book. Break away halters or basic leather halters are wonderful for horses who wear them while turned out because the leather will break if the horse gets hung up. Now if money isnt an issue then a pretty fancy leather an silver show halter is always nice but you best be rich lol I have not seen one at a store for less then $800...so there isnt a certain type of best halter made from a best material...it all depends on what you want it for.
I don't leave halters on when my horses are turned out. BUT....if I did, I would ONLY use a breakaway halter - personally, I like leather - but, there are nylon halters with a leather crown piece that would break if a horse got hung up in something.
Personally, I never turn my horses out with a halter - even a breakaway halter. What I do is to train my horses come to me when I whistle or to at least allow me to approach with a halter/lead line without running away. It is easier if you already have a horse that you've trained so that when you call, the trained horse will come and the new one usually follows.
I always go out into the pasture with a halter over my shoulder. Sometimes it's just to approach them, give them a treat, and let them go. Sometimes I'll halter them, pet them a little, and let them go. They come to see the halter as part of me and not as a clue that they are going to have to work.
I used to use leather halters but now I only use a rope halter/lead. The only reason I have a halter is to lead or tie my horse; I don't use one for show or turnout. (I have a nice collection of leather and web halters purchased over the years that are there only for display)
Other than the first day we got my new horse (we were afraid the other horse would pick a fight) we don't turn ours out in halters. Our horses are trained that if they have a halter of any variety on that they are "working" and aren't allowed to be grazing or wandering off.
We use nylon halters usually, easy on/off and we have different colors for each horse so we can grab the right one in a hurry if needed. If we take my horse off the property, we usually keep the halter on under the bridle for just in-case since she's still somewhat green. We have a knotted rope halter that my sister-in-law rides my horse in as part of her training.
I bought a nylon halter that has leather patches with Swarovski Crystals on it because it was really cheap (tack shop didn't know why they had it and made up a silly price when I asked) but it's never been used. It's really pretty so we don't want to get it dirty!