Mulch will wash out if the bed is higher than its surroundings (like a sidewalk or yard). You can trench the edges of the bed to create a depression where the mulch can go if it starts to move due to rain/wind/whatever. As far as color of mulch, I'd go with brown.

Get undyed mulch. It's brown (ask for undyed, some browns maybe dyed). Why anyone needs to dye mulch is a mystery to me. Plant a few more shrubs, so you need less of it.

Phlox subulata will fill your beds nicely and be pretty this time of year.

To really fix the problem I’d take out the paving stones and widen the bed. Then put the path back further from the house wall. I’d also lose the wiggly edge and the stone edging on the bed to the left of the door. Make everything more generous and less fussy. Then plant more plants.

Plant ground cover plants that will act as mulch. Dyed mulch just leaches out, and isn't good for plantings anyway.