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.

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