Where do your buildings sit relative to the ground? The bottom of your house should/might have a "sill plate" that marks the essential bottom of the house. Below that there will be a slab or other base that is designed to be buried underground. From the sill plate down, the soil should not start until four inches below the bottom of the sill plate. (the sill plate sticks out from the lower base of the house by 3/4 inch or so all around). Then your soil should slant away from all the buildings. If you have that, then your yard is properly designed and your neighbor's is not. It could be he has fallen into the trap of "throwing a little topsoil" on it every year. I've seen build up of 8 inches from this practice. You might talk (nicely) to him about that to get to the root of the problem. It doesn't seem like a quarter inch of soil could hurt anything but over time it can mess with yours and everyone elses drainage. Then you might have to have a professional come in to regrade the yard so you have proper drainage. Where it goes might be up to the local city planning department. On your tree, when you look at the ground now do you see where the roots flare out away from the trunk? If so then the tree is sitting properly in the soil. If you do not see the root flare and it looks more like a telephone pole sticking out of the ground, then you should dig down to expose the flare of the roots. Trunk bark is different from root bark. Trunk bark can get fungal diseases that root bark cannot. If you bury the trunk, even only one inch, that trunk bark can get a disease that girdles and kills the tree. If you are already to low with the tree, you can dig it out with water pressure from a hose and place a physical base around the tree. That would be to keep the soil from washing back in. Make it about 4 inches or whatever brings you to grade and encircle the tree completely. You don't have to go way far out into the yard with it. You only have to keep the trunk bark from staying soggy. Misplacing a tree in the ground is a very common mistake. You can even buy new trees from the garden center that are planted too deep in their own pots.

How often do you water and for how long? Could you dig a trench or a swale to let that water go somewhere? Is that area sunny or shady? One of the problems with a low spot is if the area stays under water or completely waterlogged, the beneficial fungi that normally live in the soil will die off and you will end up with a swamp type of soil. Then all you can grow is swamp grasses (including nutgrass). You might try spraying the area with baby shampoo every 2 weeks for the next two months. Baby shampoo will allow the water to penetrate deeper into the soil despite the loss of beneficial fungi. Another possible spray would be molasses. Molasses (and any sugar) is also a "wetting agent" that might help with water penetration. It also feeds soil bacteria which will help reestablish the soil fungi. You still might need to put in some cure for the standing water. My first option would be the swale and if that does not work, then a French drain.

Have you thought about buying several large bags of pete moss, and about a ton of sand. Spread this over the affected areas and use a good tiller and till this into the existing soil to a 12 inch depth. I did this in a previous home. Worked well.

I'm not sure if I really want a Rain Garden, but as rexanne said, if all else fails -- I need a plan of something I can do/try. I choose to avoid fighting with the city early on, and the city really did help me out a lot in geting my larger drainage issues solved with several other parties that probably would not have helped me otherwise. They gave me power when I would have had little to none otherwise, other than paying a lawyer and hoping to recoup the costs later in a settlement. The city still has leverage with builders in other ways (even beyond permits) in that the final handover of all the builder development owned/maintained drains, sewers ... has to be approved by the city before handover occurs. So the builder pays maintenance on all stuff till its good enough for the city to take over and becomes property of the utilities dept. But if the city tells me to live with it ... then all I can do is whine and take it to a political level and whine to city council, etc. So far, they have been pretty good and actually wanted the best for me. I cooperated on this project and made is easy on them, as they could have dealt with 5 difference property owners instead ... so dealing with just me was easier. Besides a rain garden, I can only envision putting in a drain to the underground sewer (costly, but very effective) or some kind of all-season, low volume pump (like a sump or pond pump) ... but would be potentially problematic and would require electric runs throught the yard and into the wet zone ... not exactly appealing either. This is all assuming I can get them to pay for this without spending an arm and leg myself.