Think about your grass the way a farmer would his paddock - you want green, weed free healthy grass with no chemical residue. if it's been sprayed you stay off it for a while, after it's been cut/foraged you give it a rest and time to grow. (Which doesn't happen with lawn since people like it short) Same as if you were looking for herbs in the shop or garden.
Pee isn't much of an issue after rain, it's mostly nitrogen and phosphorus and will break down and be converted pretty fast. You can water your vegie garden with diluted or undiluted human pee as long as the pee-er isn't sick or on medication. Animal pee isn't that much different. Once it's rained a few times it's mostly gone. Horses avoid the grass patch they pee on for a while, but they pee a bucketload in one spot. Guinea pigs will probably just not eat the grass if it smells weird.
Petrol lawnmowers leave a really small gas residue on the leaves, ideally you'd cut grass for the pigs before you mowed when it's longer anyway. I just grab handfuls of long grass but it's pasture grass, so that's different. No mowing happening at all. If you could leave a small section unmowed and trim with hedgeclippers/scissors/whatever you have, that'd work. Actual lawn clippings after mowing are smushed up and limp, wouldn't feed them those.
If you want to add some leaves, try adding a few herbs/vegies to the garden - coriander, lemon balm, lemon grass, parsley, leaf celery, kale, to start. Have a look at the nutrition charts and see what works for you. You only need to add in a few leaves a day or week to perk the pigs up.