Violets and Glechoma hederaceae. I can’t help with the grass question. If it were my garden I’d mow them and call it a lawn.

The non-herbicidal method of removal would be manual - weeding the plants out. Good luck with that on a 1/4 acre!!

I find the obsessive focus on pristine, weed-free lawns in this day and age of climate change and our continual messing about with the environment somewhat disconcerting. My own lawn (and I use that term broadly) is a mix of turf grass, weeds and wildflowers. And some moss. It gets mowed like a traditional lawn but is never fertilized or watered yet still remains green through our very dry, droughty summer. And blooms very attractively periodically :-) Currently I am finding random patches of hardy cyclamen blooming!!

I planted some sprigs of Blue Star Creeper, Isotoma fluviatilis, in my 'lawn' this year.

Did you sneak by my house and post a picture of my yard? A large part of my lot looks like that. When we moved in, I had delusions of a beautiful green lawn but since I am not interested in either adding water or chemical weed killer, it isn't going to happen. My approach is to add more interesting plants and divert the eye to my perennials. I do have a large area in the front where ajuga pants have crowded out most of the other plants and that looks good to me.