I don't know how I missed this the first time around. You'd probably have better luck if you seeded in the fall. Cool season grass seeded in the spring has a tough time making it through the summer. Dutch White Clover may be a good option for you. It's drought and heat tolerant and used to be included in grass blends until the weed and feed products started getting popular. If you have alkaline soil, strawberry clover may work better than Dutch White Clover.

There are various listings of groundcovers suitable for the Ohio area, typically provided by your local extension service. I've included one for you to review. You will need to determine sun/shade suitability for your specific needs. I wouldn't be overly concerned about toxicity - way more plants are toxic to some degree or another than those that are not. The cats are not likely to bother them at all and now is a good time to start training your child not to put anything in their mouth that mom or dad did not OK first! Besides, unless it produces berries or some sort of fruit, munching on garden plants is not all that appealing to kids either - just try and get them to eat their veggies :-)) Here is a link that might be useful: groundcovers for Ohio

I've researched Leptinella quite a bit. It is from New Zealand. Everything I have read suggests it will be fine in full sun as long as it stays reasonably moist. It should stay evergreen year round in your zone, but it doesn't reliably do so in Zone 9. Platt's Black is a very interesting cultivar. You might want to try a small amount to see how it does. I am unsure of its toxicity but I doubt it is non-toxic (the vast majority of plants are toxic if consumed in large amounts). It probably will feel great to walk on, though. The mint advice is definitely right on -- full sun is a no go. The thyme I believe has smallish flowers, so in theory they wouldn't be huge bee magnets. I do not have experience with it though, so I would recommend doing some more research on it. Mazus does need some moisture. Though it prefers some afternoon shade, it would probably (like Leptinella) do okay in full sun if you water vigilantly. I doubt that Mazus is non-toxic. I know the blue star is reported to be a skin irritant (I've touched it a great deal and never had an issue.) It is also toxic if consumed in large amounts, like most plants but I am not aware of it being particularly toxic (like carolina jessamine where only a flower or two can kill a small child or dog). There are few plants that are truly non-toxic -- mostly true herbs. So, if you are worried, I would go with the thyme. That said, I doubt either your kids or dogs would be able to put away enough blue star or leptinella to cause a problem.

Co-mingling groundcovers is not usually a very successful practice :-) One always tends to be more aggressive than another and will claim most of the territory, if not overwhelming its partner completely. While I tend to agree with Mike about the sweet woodruff - a nasty piece of work in my book - if you are happy with it, I'd just encourage it to spread more (I can't believe I said that!!!). Otherwise, I'd pick something else altogether. Vinca minor is a reasonable choice but may need some containment in certain areas. My favored shade groundcover is Adiantum venustum, Himalayan maidenhair fern. Evergreen, attractive and spreads moderately. Also a Great Plants Pick.