You have several issues preventing you from having a healthy, full lawn. The first is the dogs. Grass can tolerate a certain amount of traffic, but having many dogs on it ever day doing what dogs do will thin out the stand. The drainage issue is also a problem as wet areas don't allow grass roots to breathe properly. Another issue is establishing grass with dogs. I see you are in NY, which means you should use a cool season grass. Cool season grasses are best planted in late summer/early fall and require no traffic for at least 8 weeks, and if you use Kentucky bluegrass, it would be wise to stay off it for longer. Laying sod in the late summer would give you an instant lawn, but again, you would have to stay off it for a while, but it would be ready for use in the spring. If the area is large enough you could seed half of it, and keep it off limits with a temporary fence for a year, then do the other half. Not an ideal situation, but if you want grass, you will have to make compromises. The mulch idea is probably the easiest way to go, and requires no care, just occassion redistributing areas that thin.