If you mentioned the goldens at the outset I would've stopped reading and said "there's your problem." My little 40 pound pointer is shredding the flat parts of my yard that stay damp all by herself.

The tall trees are what they are. You're within your rights to prune any branches that cross the plane of your property line, but making a shallow-rooted garbage tree like a Leyland lopsided and MORE unstable wouldn't be my first choice. The mud is a whole other issue. If you're getting mud washouts it's worth a conversation with the town or county building office. They may just be required to put up a silt fence, which won't be the most attractive thing ever, but at least it'll stop the mud.

The water issues are impossible to talk about with seeing the lay of the land.

"This results in a veritable mudslide into my yard..." Got a picture to show this condition? (The description could mean different things to different people.) If there was a substantial infiltration of mud into your yard, you MAY have legal recourse. If it is a small amount, the legal route may not be at all practical. Pictures could help determine what a practical solution might be.

As far as shade from Leylands, you'd need to change to a plant that tolerates the shade and wet conditions, which is probably some kind of low groundcover. Zoysia will tolerate some shade, but not poorly drained soil.

An inexpensive solution for the flow of surface water and small amounts of mud runoff that flow into your yard would be to install an erosion control tootsie roll / bale or a v-ditch.

You have no recourse for the shade.

If you are truly getting 'mudslides' into your property then you definitely have legal recourse but know that anything other than small claims court is extremely expensive and you need to weigh the financial benefits of either.

Well I would try to make lemonade out of lemons. Yes you can prune, or "limb up" the branches that hang over onto your yard, but you wouldn't want it to look hacked up. You may not be a gardener and so this solution may not appeal to you, but it really is not hard to put in a mixed shrub border (very low maintenance, really!) back there, with some large stones or small boulders mixed in, buried 1/3 or 1/2 to look natural. Then you just need leaves raked in there, your grass clippings, and pine straw if you have that, to use as mulch in between the mixed shrubs. This really will catch the mud and water and prevent further erosion. The shrubs should be evergreen, mostly, so it will look nice all year, and with varied height and foliage, and shade tolerant. You can search for that and it is easy to find, or a local garden center can help you. If you plant native shrubs you will probably do even better. This mixed shrub border will give you something very beautiful to look at, attract birds etc, the dogs should love to explore in there and not make a mess. I have one of these at the back of my yard, I make a small walkway behind the shrub border in case I have to get in there, I layer leaves and small branches and pour the stuff from my paper shredder back there (you can't see it) and it makes a nice path. I have cut and pasted 2 examples, it can be much simpler than the below examples, but use a few different shrubs, not all one kind. You don't even need to put in flowers or hostas or anything other than the shrubs. In my vast gardening experience, shrubs are the easiest of all plants. You are in a great gardening area, Atlanta, to achieve this.

http://www.bing.com/images/results.aspx?q=mixed+shrub+border+with+stone+outcroppings&form=MSNH14&refig=0df17a0772a1474cbbc125bffcc81840&pq=mixed+shrub+border+with+stone+outcroppings&sc=0-19&sp=-1&qs=n&sk=#view=detail&id=32DCD70D6EEE49DAB1A040EC510D3E798E4F48F4&selectedIndex=85

http://www.bing.com/images/results.aspx?q=mixed+shrub+border+with+stone++&form=MSNH14&refig=0df17a0772a1474cbbc125bffcc81840&pq=mixed+shrub+border+with+stone+&sc=0-19&sp=-1&qs=n&sk=#view=detail&id=251E31D0E7653476CBE29DD3A8F04EEE12DCE152&selectedIndex=229

cthen,
For the side that is slanted down towards your house, go out this spring and buy yourself ornamental grasses.
Make sure you water them this summer, they need water the first year you plan them when it gets hot.
They will get big, and beautiful, look up some online to see which ones you like. They are fantastic for erosion!
For the yard area that is shaded by the leylands, I would plant either a ground cover there, or find a grass that tolerates shade. You may have to search to find a shade tolerant lawn grass. The drainage is bad there, so you should buy some composted cow manure and till it in before you plant any seed there, but since nothing is growing there anyway, you mine as well. Make sure you
level the ground out as much as you can with a trowel before you seed.

Why wouldn't you plant huge gorgeous ferns, like Ostrich or Cinnamon in the wet, shady area? And perhaps something like Cephalanthus, button bush. If wet and shady is what you have, than plant stuff that loves it.