Need to fix the grading of the backyard: remove the sod or turn it over

I'm not sure how much "fair" applies to geology. If you have the downhill lot, gravity is just going to do what gravity does. :-)

It looks like water is definitely draining off of both yours and your neighbor's lots into the depression that the fence sits in. If the soil can absorb all the water quickly enough, there's no issue. Your neighbor's hard surface parking lot, and your hard surface patio, both reduce the amount of soil available for water to absorb into and cause more surface water to flow wherever it flow. Very little water will absorb into the ground under those hard surfaces. (Engineering for this kind of run-off is a serious engineering issue when facilities are built with large parking lots and large roof areas).

If your fence posts are rotting at the base, be sure you replace them with pressure-treated posts. Maybe the original posts were not pressure treated? If not, even modest amounts of moisture would rot them fairly quickly.

The same principles apply to water that seeps into the ground and migrates in your direction. You're further downhill. There's just no getting around that. Once the ground is saturated, the water will pool up on top and rapidly flow downhill. And of course if there is more high ground on your neighbor's side of the fence, gravity is going to tend to move the underground water in your direction. You definitely want good drainage away from the foundation of your house. Otherwise, you'll eventually end up with seeping basement walls, or a crawlspace you could use a canoe in.

Is the planned dry well the only option you have for drainage? Is there a slope in any direction that would let you run a perforated 4" pipe (ABC or PVC) to the street to drain excess into the city storm drains (essentially extending the french drain)? The blue line in your diagram shows water flowing away from the house and garages toward what I presume is the back of your lot, but you don't specify what's back in that direction (an alley? Another lot line? A street with gutters and storm drains? A field?).

Generally speaking, though, if you have those french drains built right, with perforated pipe buried in gravel, you can tie them together with additional underground pipe sloping downhill away from the house. If you bury them in plenty of sand and gravel, that will create a lot of additional absorption for water, and provide extra time for the water to absorb into the surrounding soil.

I'm curious if you are on city sewer, or if you have a septic system and a drain field. If you have room in the back of your lot, and if drainage is really that big an issue with water pooling around your house on a regular basis, you could even potentially build something like a septic system drain field--without the septic tank--to direct more water away from your foundations and give it more underground area to absorb into the soil. But that's kind of the big issue. The ground can only absorb so much water in a given amount of time. That varies widely based on the composition of the soil. Dense clay soils can't absorb as much water. Soil that is already saturated can't absorb more water. So the water is going to run somewhere unless you can spread it out underground and expose it to a bigger area of soil. Even then, if the flow is too high, those french drains will back up and you'll have water pooling up above them.

I worked as a design tech for civil engineers once upon a time, and I remember stories of a city sewer system that was under-designed to the extent that when a "100 year storm" would occur, water pressure would back up and water would flow back up out of manholes, with the manhole covers literally pushed up off the ground riding on top of the water flow out (that's a lot of pressure).

Only re-grade if you're getting water intrusion or even a damp musty basement condition. If neither of those yet, then don't do anything. Otherwise, re-grading is a good deal of work, but quite simple in procedure with just a shovel.

Dig a shovel depth trench at least 6-feet away from the house that pitches down toward the street or into the back yard. What comes out of the trench goes against the house & then you widen the trench toward the house while digging more & more shallow to achieve pitching away from the house.

The Gas Meter (may be the picture angle), Front & Driveway sides all look good. The patio should be ripped out & redone with a properly pitched concrete slab, re-grading would be part of that job by default. But really, only the AC's side would need to be re-graded at any point or in your lifetime. But, put elbows on your gutter downspouts, they aren't helping matters at all.