As for tails one of their main purposes undeniably is communication, and yeah they incidentally are used to shield the genitals, but I don't think you can then say they definitely don't aide in balance while running. I think to a degree they probably do. Greyhounds have longer tails than most dogs, my current dog which is 25% greyhound (at least, has thrown very sighthoundy) noticeably has the longest tail I've ever had on a dog of mine. I don't think the point about deers not having a long tail really holds, they're not related to dogs at all and obviously their ancestral lineage never evolved to use the tail for balance, just incidentally it didn't happen and they made different adaptations for running. More significantly as you say the cheetah does use it's tail for balance and cheetahs are carnivorans and so closer related to dogs and perhaps a better clue. I don't think it's going to make a huge difference with the tail cut off half way or whatever, and I doubt it matters at all for close-quarter altercations and even catch dogs running up to grab onto pigs. But for coursing dogs that chase jinking and jiving prey I think it would play a role. I mean dogs do turn their tails in certain ways during chases, they don't just loosely flails behind them.