Pocket gophers are called vermin for a reason: they are NOT harmless. Those who do not garden within the range of the Pocket Gopher need not worry themselves about the survival of these vermin. I personally supply them with 15 acres of prime habitat. They can leave three quarters of an acre to my garden. They ruin lawns, they eat roots of many plants, not just ornamentals, they chew and girdle trees and shrubs, they chew holes in irrigation pipe, their tunnel systems destabilize hillsides, dams and dikes, their burrows can lame or cause fatal injuries to horses, they destroy young trees planted for reforestation of areas destroyed by wildfire, and they harbor ticks that carry Lyme Disease. They have their place, as they process enormous quantities of soil when they tunnel and they feed raptors and predators. Too bad they don't tunnel in rocky soil. They show up after the soil has been improved. One of the serious problem they cause is secondary poisoning. Gardeners and farmers suffering losses from pocket gophers often turn to poisons that, if the dead gopher is ingested by domestic pets or wild raptors, cause death. The predation of my dogs on pocket gophers is a substitute for the predation of native predators excluded by my fencing. Long live the ratter.

I am generally opposed to using chemicals except when necessary. I did buy some weed-killer that says it's specifically for "woody vines" or something like that. After pulling up/out as much ivy as possible, I am carefully painting the weed killer on the new leaves when they appear. It seems to be working, as I have not seen much new growth. On one rock retaining wall, I also painted the weed killer on the cut stems when I couldn't pull them out. That seemed to be very effective. Roundup will work and is supposed to be relatively environmentally benign. Everything is relative, of course. It is much more environmentally friendly to just pull off the new sprouts when they appear. It will take longer, but eventually the plants will die if they don't have leaves to make food. I agree we should make ivy illegal.

There are about three thousand species in the genus that contains the insect commonly known as the lady bug (which isn't actually a bug at all, since it possesses two pairs of wings, a hard exoskeletal wing and a membranous flying wing). I am by no means as expert in beetles, or insects, but as far as I know, the genus does not feed on plants. If any do, the damage would be visible as chewing or notching on leaves.

Sounds like you'll have to renovate in the fall if you want a nice lawn. I'd leave it be for now, mow the weeds and prepare to kill it all with glyphosate come August in preparation for complete renovation. You MUST apply pre-emergent next spring if you go this route, as the weeds will produce a lot of seed.