Depending on the severity of the runoff you might want to contact a Civil Engineer. As an engineer myself, with a specialty in drainage and hydrologythere are many issue to consider when dealing with offsite drainage. For instance if you redirect it on to another property it's possible you could be held liable for any impacts tha occur. Just a thought.

We agree with Geoff's suggestion and would also recommend speaking with a local landscape architect. They not only handle plant specifications, but also deal with drainage and irrigation. They will be able to assist you or point you in the right direction to solve your issue.  

A landcaping contractor might help. Depending on your location, you may be able to find someone here: http://www.guildquality.com/find/Landscaping-and-hardscaping/North-America

You can install a flexible downspout extension on your downspout that is next to your front step and run it behind your bush out past your mulch to keep the water from washing your mulch away.  You can buy the flexible downspout extensions at Lowes or Home Depot.

We keep a spray bottle filled with the solution below on the counter and use it as needed. We apply sealer once a year.

•1/2 cup rubbing alcohol
•2 cups water
•8 drops dawn dish soap

What is the water source? It looks like it is coming from a drain pipe behind a shrub? I would recommend relocating that to dump to a different location. No matter what type of edging you install on the bed with the mulch, a massive amount of water from that down pipe will cause some overflow and mess. Maybe you can incorporate a stone trail where the water flows, if you can't relocate the down pipe. Ideally, I'd want to see the drain pipe go under the side walk, and out by the grass, or if its by a garage, down the driveway.

Using warm soapy water is the best for daily cleaning.  There are many granite cleaners on the market that will help you if you really want to scrub and bring back that original shine.  Laticrete (formerly DuPont) stone care products are great.  They make a revitilizer spray that is excellent for weekly cleaning.  It has a bit of sealer in it as well, so it will help keep those tops shiny.

Emerald Green Arborvitae

Install a decorative stone border or edging that will help hold the mulch in place.

Warm soappy water! Cleaning products takes away your granite sealer.

We use and reccommend TS&G Disinfectant & Cleaner. It is made by MIRACLE Sealants Company and you can use it safely on Tile, Stone, & Grout. It kills Molds & Mildew, Disinfects, Bacterial, Fungicidal, Cleanses, Deoderizes, & Eliminates Oders.

You night try Pine straw instead of wood mulch. It's less prone to washing.

Try a dwarf arborvitae. They look similar but don't attain the height of a leyland cypress. 

In Georgia a wall retaining over 48" grade requires an engineers stamp.

You might consider terracing and have two or more lower retaining walls. Be aware of hydrostatic pressure and provide adequate weepers to deminish concern for hydrostatic pressure. Good Luck!