The water table around my house is very high with a lot of surface water due to clay subsoil. My experience of 25 years in my current house (boat?) is adding a french drain around the house is a patch to a larger yard grading problem. I've done this and eventually they clog and need maintenance. Over time I've regraded most of my yard so water flows away from the foundation. Gravity never fails and never clogs! My guess is your patio is sloped incorrectly towards your house. Short of replacing the patio, a french drain is the only alternative I can think of. They work when installed correctly.
I just replaced my patio this summer. We had a surveyor transit, built up the area with dirt/aggregate and put in a slightly steeper pitch. The one small part slopes towards a french drain to my downspout. I was afraid the water would run into my neighbors yard and I don't need those problems! The larger section pitches towards a special garden area that was dug out and filled with mulch and water loving plants. It gets very soggy! Again, my approach is to slope the water away from my foundation and then do something with it there. Through hard knocks I've learned solving a water problem at the foundation is the least desirable place to address the problem. Unfortunately for many, prior mistakes make it the only alternative.
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).
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I would add the new drain at the edge of the grass up to the right (not two close to the trees) and continue it to the original drains. You can tell the original drain is working because it is mushie beyond that point this would have been the location I would have used at first but it is longer thus cost more and the previous owners may not have cared about the lawn much in the winter. A new drain should do the work of drying out the lawn.