@wildflowers: I really have no idea what kind of grass that is, some mix of pasture grass is my best guess. When we bought this property over 30 years ago what is now our yard and gardens was being used as a horse pasture. We've never formally seeded a lawn, just mowed what was existing. Lots of dandelions and clover, various types of grasses. You've rather inspired me to try to identify what grasses I do have -- in our outlying fields (same mix), we let the grass go to seed and it gets about 3' tall with a pretty spray of seeds. I will make a point to photograph and try to ID them.
Re maintenance of the herb garden: I edge in the spring using a square edged spade and a string line (this garden is laid out as a labyrinth so I like to keep it uniform), and top-dress fairly heavily with a local compost blend. Then I mow the paths throughout the season on about a weekly basis (more often in early spring). My husband also hits the edges up with the weed wacker now and again when they get unruly, although I don't care for that look as well, it's a bit rough and uneven. But, clean is clean and by then the plants have mostly softened the edges anyway, so I don't complain. Sometimes I will re-edge in the fall before laying on a thick layer of leaf mulch, but usually not.
