The first picture above is creeping "phlox", not flax. Phlox subulata is the name and it is a native plant that is usually available in the spring in stores but not much during other times of the year.

I second esh's caution about periwinkle- i planted it on our hillside and have since decided to go more native in that area and now am faced with pulling it all out. It has spread far beyond what i wanted. It can handle full sun and most likely the pugs once it was established. Same goes for ivy- it turns into a real problem before long. Really, you have to be careful with almost any groundcover- the qualities that make them good groundcovers can also make them get out of hand fast. Just knowing that is half the battle, though. If you plant something native (like ajuga), then at least you don't have to feel bad if it gets rambunctious. I tolerate to actively encourage creeping charlie in my back yard as an alternative to grass (it's nonnative, but endemic). Another native that could probably take the abuse is virginia creeper. It has nice fall color. It climbs, but on a slope would crawl. Daylilies may be too grass like but would give you flowers and do well at stabilizing slopes. They aren't evergreen, though. Phlox is made for that situation- except i don't know that it could hold up to dog abuse. Same goes for sedums. Vetches are used up north for stabilizing slopes along highways but i've not seen them used that way down here. Not sure if they don't like the heat or what. They are very tough. If your hub doesn't like grass you could try clover. Or a mix of clover and grass.

puddle, please don't stop posting just because you don't know the names for things! The only way to learn is to post and be corrected. Goodness knows, i often remember one or 2 syllables of a latin name and sort of fill in the rest from memory- almost never correctly (and get corrected a lot ;). So you're in good company. Triangle John's been known to do the same... LOL

eleagnus angustifolius aka 'sweet olive' 'Russian olive' is potentially invasive.

eleagnus umbellata aka 'autumn olive' is potentially invasive.