How deep in ground should I install 4x4 PT fence posts?
If you already bought (and can't return) 8 foot wooden posts and you want a 7 foot high fence (or "at least 7 feet") I would suggest not burying them at all, as that's doomed to failure. Put metal post bases (or metal posts) in the ground and bolt your wooden posts to them. Otherwise buy 10 or 12 foot wooden posts if you want 7-8 feet above ground and you want them to stay put. Or, bury them 3-4 feet deep and bolt 2x2 or 2x4 lumber to them aboveground to get to 7 feet (or more) above ground level.

If you didn't already buy them I suggest you learn about metal posts, they are usually much less expensive and easier to set, as well as somewhat safer around livestock than PT wood. But you still need enough post in the ground to hold the post in the air stable, and unless you are drilling holes in solid rock to place your posts in, a 1-7 ratio is dubious in the extreme - and so is placing concrete footings above frost line. In reasonably solid ground an 8 foot post can be driven in 2 feet and leave 6 feet above ground, but your corner posts will definitely need some bracing to stay put, and/or will need to be longer.

Wire (mesh or not) is actually more demanding on the posts than a wooden fence - the wire pulls and provides no support - wooden fence sections provide some bracing to the posts.