I have only done this once, in my own backyard, but I am also in the business of aesthetics, so I thought I would chime in. In my situation we went with option #2.
I would not recommend option #3, due to purely aesthetic reasons, unless you would like to keep the fence low all the way down for some reason (light, neighbors wishes, view). Having the gate suddenly jump up by 4' at the end would be quite strange looking, in my opinion.
If you go with option #2 the fence will be high in the back corner, which will allow for taller plants and more landscaping options, and will just look better in general. You get the same effect with option #1, however, it is more of a pain to do all the cuts necessary to slope the bottom, which after a little bit of growth from the landscaping you will never see. It would really be a pointless detail. In my experience, once we got building we were very happy that we had decided against doing that very thing- it's enough work already! And after just a couple months of growth the bottom of the fence was totally obscured anyway.
I wonder if your house is a mid century style? Horizontal fences always look great with this style of house- I also prefer them over vertical for almost everything. I wish I could send you a pic, but I don't live in the house anymore and am renting it out. Anyway, I hope my two cents helps you decide!
Before you start, talk to your neighbor first, get a feel for any concerns if any they have. Then inspect the post to see if it would hold a section of fence. Then go to it.
EDIT: Ok lets try a different approach, dose your property line split the post, (red line), or meets it (green line)?
This will make a difference. I would also consider placing a stationary panel first because the swing of you gate would be hitting the curb on the driveway.
Best Answer
Take a look at the category known as 'hidden deck fasteners'. It's typically a biscuit slot with a metal or plastic fastener that keys into the slot and then attaches to a post. (Or in the case of a deck, the joist.)
Top tip if you want to save time and wear on the biscuit joiner: have your lumber supplier groove both sides of your planks full length.