And any thoughts on identification of the tall grasses around the unit in this photo? Much appreciated

(Landlord's terrible electrical wiring is another conversation 😅)

Just about any small to medium sized shrub would work or maybe a big perennial. Just for grins and giggles how about:
Boxwood- try a variegated one for fun
Euonymus
Rhododendron- pick a color, any color
Rose- just don't plant it too close
Chamaecyparis- lots of different options
Peony
Day lily- there are some big ones out there
Viburnum
Your grass is Miscanthus, which one I don't know, but some of them can get quite large especially if you give them lawn food

Leave at least 2-3 feet around the ac unit free of plants

What do folks think about planting a mix of climbing and oakleaf hydrangea in this corner? Looking for a simple way to fill the area, and that combo seems to fit the bill.

Like Rj said, don't be planting anything close to that a/c. It needs to breathe!