You have provided no photos, sketches or any other means for us to evaluate the property. Without these to give us a basis for comment, there's not a lot to say. Except that two big, active dogs and a healthy lawn just do not go together. We also need to know where in the world you might be located....that makes a big difference.

Ditto on photos. Sounds like your starting point is addressing the drainage issues.

If you have ledge under a thin layer of topsoil, then you have problems I can't solve. Otherwise, I'd put in a dry creek bed, basically a channel for the water.

Birches can absorb lots of water. We planted one in a damp meadow and the lawn around the birch stopped being so soggy in the spring as the birch grew bigger.

A tree or two would improve your landscaping, so I'd plant a birch or a clump of birches in a corner.

Soggy ground should be good for growing grass, but sog and lots of traffic (do your dogs run around in the yard) is bad until you have lush grass.

Im not sure why just raising up the low spots (ie where the puddles are) wouldnt solve the problem .

I would just bury the whole yard in a truckload of woodchips- thick layer, at least 6 inches, a foot is even better. Seriously! At least youd have a useable yard. It mellows to a nice color and compacts down and becomes a comfortable surface to walk on. And over time breaks down into organic matter that enriches the soil.

THen, you can take your time to decide where you would put trees, raised beds, walkways and patio or deck living areas. Grass and several big healthy dogs playing, pooping, and peeing are not really compatible, so you may just want to stick with wood chips as your ground cover and decide to enjoy your dogs instead.

Some cities give away truckloads of chips from their tree trimming operation - arborists & tree cutters sometimes do as well.

You would not want to LEVEL the yard, but pitch it in a specific direction (at a minimum of about 2" fall per 10' of run) so as it can drain off, and also fill any depressions. Make sure you calculate it out first so that you're not trapping water at the uphill end, which I presume is at the house.

It'll be hard to have a lawn with dogs. You might want to settle for the mulch but 6" sounds excessive to me. I'd settle for more like 3".

I agree with the above. While it is not clear in the pictures, it looks like the area behind the fence is lower that your yard. (based on the trees in the background)

If this is so it would be a simple task to add swales so the water goes down the slope and does not puddle in your yard. They only need to be a couple of inches lower that the surrounding yard, and can be created by lifting the sod, removing the dirt underneath, and relaying the sod

yes the entire backyard is fenced in.

Train them to go in a certain area - whether actual pee pads or just a designated spot. DOgs are usually pretty smart and trainable - not always, but usually.

If you only want to do 3 inches of mulch and dont mind weeds coming up in a matter of days, then go ahead, that's an option. I would do the grading of the yard first (yes slanting away from the house is good) and then lay down the chips.

You can’t just grade your yard to drain the water off if it’s going to cause a problem for your neighbor who is downhill of your water issues. Get a landscaper who is experienced in drainage work to look at it make a recommendation. If you can redirect the water to one specific area of the yard, you might be able to make a rain garden in the low spot.

He should consider the neighbor when determining the flow of the water from his yard. However, I believe the legally he has a right to direct the water down the natural slope. Also a neighbor can not block the natural flow.

The neighbors in the neighborhood should work together to make sure everyone's yard drains properly. Water can damage structures like houses, etc. but also excess water can damage grass, trees, and bushes; and be a breeding ground for mosquitoes, some snakes, and similar pest.

i do not have any neighbors behind my house- only woods and a drop off. But i do have one neighbor on the side of my house.