You can certainly treat her, and in your case I would since you are depending on her for house milk. It's really nice that she isn't showing a temp or a hard quarter. The antibiotic treatment has a chance to do a good job.
What you saw before, sack with liverish looking spots, was the placenta. The liverish spots are the cotyledons that transfer the nourishment to the calf from the walls of the uterus. The glop behind her is nothing to worry about. Cows will discharge for awhile after calving as the uterus involutes. If you see a creamy coloured non-transparent discharge, that would be pus and a sign of infection, but normal run of the mill mucus with some blood in it is perfectly normal. Cows don't get infections in the uterus too easily, and if it was a normal birth (no vet work, no rearranging the calf, etc) with nothing to introduce bacteria into the uterus you won't see them often.
I would bet the reason she has mastitis in the quarter that leaked is *because* it leaks. Easy milking quarters are much easier to infect with environmental dirt (you know what I mean by that) because when she squirts out milk the teat canal is open, and when the milk stops squirting, it takes awhile to close up again. All the time it's open it's available for bacteria to get in there. Not a whole lot you can do about cows like that except to keep the environment clean where she lays down. Commercially, cows are dipped with disinfectant after milking and ideally go out to eat at a feed bunk or pasture, so they don't lay down until sometime after they are milked. The teat canal will have closed by then and the chance of mastitis is lessened.
From what you say you cow is normal and healthy with a case of mastitis that should clear up with treatment and/or with the constant milking you say you are doing now. Do NOT expect the calf to clean up that quarter, btw. I've heard people say over and over a calf will clean up a cow, but that's only if the calf is pretty big and needing a lot of milk, that is, a month or two old. A baby will not take much for awhile and as you noticed, that quarter isn't tasting so good noe. The calf will avoid it since it has three other good faucets to work on. So go after that quarter and get it cleared up and she should be good to go.
Jennifer