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.

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.

Emerald Green Arborvitae

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

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. 

As a builder I could tell you there is no right answer. I built my home on the water and I have switched from eal grass to artificial . My reasoning was to conserve water and time. Now I can also focas my time on other things. I've addded a complet creek that runs down my entite property into the lake and addded palm tees and other floweers and tees. The amazing thing is hoe green my yard always appears from lake and now I have installed over 20 other homes with the same product. For me the switch was completely worth it as well as all the new clients I've gathered from it. It's a personal decision for sure. www.greenbuiltwa.com

1. Do not think about doing it yourself.

2. Check with a treee removal professional

3. Check with the city in which you live and inquire if there are any restrictions on the size of tree that can be removed.

4. Get two or three prices

5. Check the location of the utities and clearly mark them.

6. Day of work, if they are on your property they have to put cardboard under the their truck to catch oil leaks

7. Double check on what is "leave the work area clean" get this part very clear.

Philip Anderson

HDR Remodeling

Berkeley Ca

I'm with you guys on the eating part! We had a bupper crop this year and the freezer is full. For those areas where we don't want them, there is product we found at Southern States that works. I bought it for our poison ivy but it ended up working on other things. The name escapes me, but it come in larger bulk containers and you mix w/ water.

I would wait until the first frost, that very next day spray or paint leaves with Crossbow (if spraying isn't an option, as in beds with other plants or single shoots).  The sap in the stems will draw the herbicide down to the roots as the blackberry plant goes dormant in winter and kill it at the roots.  May have to repeat again the next year, but it has been very effective for us.    

I second the nursery, but also make sure your soil is plant friendly. I lost a pretty large investment in trees because my soil is not good for growing. Next time around I will need to condition the area I plan to plant.

Dig it up and send it to me. I love blackberries.