how do I straighten my my gate attached to the neighbors privacy fence?? I can not open the fence due to the leaning into my yard, it was working great before winter hit
Privacy fence to gate issues
fence
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Eight feet is a common distance between posts, due to it being a standard lumber length. You will have a stiffer, longer lasting fence at 6-7 ft. An integer number of pickets with your desired spacing is a good way to decide from a design standpoint.
I prefer to sink posts to at least frost level (42" in my area). The 1/3 advice is good if frost is not an issue.
I also prefer crushed gravel (21AA or paver base) to place wooden posts (versus concrete or pea gravel). 3-4" in the bottom of the hole (beneath post), continuing up the sides of the posts to just below the surface. Give the post a push after first 5" and tamp along all 4 sides as you fill. This technique will greatly extend post life and minimize rot.
I've used the gate brackets that you build into the gate which have hinges built in. They will stabilize a 42" gate (each side). Extra long screws on the top hinge are helpful. Stainless steel cable run diagonally (with adjustment turnbuckles) run from the top of the hinge to bottom of the gate (at the opposite corner) will make it sag proof.
Disclaimer 1 user of product, no financial connection: 
Disclaimer 2: Haven't used it but sounds interesting, YMMV (Near) Instant fence post backfill
A general rule (not sure if your neighbors abide to it) is to build the fence 1-2 feet inside your property line. You mention that the fence post is presumably in their property so even butting up to it would be in their yard. You best option is to ask them to join your new fence to it and get a simple contract to protect yourself. Either that or put up all four sides in your property.
You may also need to get a permit or permission from city/municipal/HOA depending on your location. Your neighbors could have even blocked you from getting a fence because of easement right between yards, all things to look up before this endeavor.
As for the height difference, it's more personal preference than anything else. Personally I'd recommend one size fence for the front of your preference and one size in the back of your preference which may or may not be the same. Like if you want front privacy but have nature in the back, it allows you to get away with 6ft in the front and only 4ft in the back. I'd mainly recommend the same size in both front sections if your yard is level otherwise it will look off from the street.
In the front you may want to fade the height difference but I wouldn't even bother in the back. Could be a nice design feature if it fades into a gate post or something.
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I would use a brace from the foundation to the top left corner of your fence with a jack - foundation won't move and use a thick plank to spread the load either end...