Well, this is not coming from a Kyo expert, not by a long shot, but here’s my take on this. I must warn you that I have the very same questions about Kyo, also - so this isn’t strictly an answer, more an opening for discussion 
Anyway. I recently tried Kyo in N. I’ve always played in C (no run), so running and short jumps in general are pretty new to me. But yeah, I found his running japs to be useful - though the same can be said for most characters in a run groove, I suspect. I think a specific advantage with Kyo is that you can cancel your punches into flaming punches, which move you closer to your opponent, and even if they attempt to counter, you’ll usually exchange at least. Always cancel your punches into flaming punches - that way you get a free follow up whether the punch is blocked or actually hits. Of course, if it hits, you can continue with the punch chain.
Then, when I’m in close I’ll use jabs at close range, forward kicks at medium, and good old s.Rh at long poke range. Like I said before I never really used his special kicks, so I’m not sure about them. Though I think the reason I stayed away from them was because they seemed to leave you vunerable if you didn’t hit (or even if they were blocked). At least, you can chain together some hits at closeish range, which will open the distance out a little more. Then you’ll be ready to use the flaming punches again to get closer. This is what I generally do with Kyo. Be careful not to get too close with the flaming punches (especially Fp) or you’ll get punished. Also, I’ll stick in a level 1 punch super (erm, the qcf, qcf P one) when I start to predict when the opponent will attempt a counter. You can do some baiting with this, also - throw in a fierce flaming punch that’s a little closer than normal, and just block the counter. Then block a little more until you can reset the fight. Next time, do the flaming punch in the same way (so, closer than normal, like a mistake), but instead of blocking rip out your super. Stuff like that works because you have a good few options at this range, and can play mind games quite easily - say by varying when you do the overhead elbow in the lp, mp punch chains, or by halting the chain prematurely, pausing, doing it again - and then also by doing the first two punches, then jab, and throw them. For me it’s all about trying to find a new pattern there to trick your opponent, or playing games with repeating the same pattern, etc, etc - you know, standard beat-em-up game mind tricks.
Obviously, if your enemy doesn’t get scared by this (in my experience they usually don’t, since they’ve played my Kyo right from when I first started using him they watched all my tactics evolve and know they can ride it out waiting for a mistake 
), they’ll usually just wait for a chance to hit and go for the knockdown, or they’ll have a good roll and time it better than your flaming punch, or they’ll jump away to long distance and force you to approach again. Each time the fight gets reset, and in my experience it’s this time where Kyo seems to fall a little short. Once your staring at each other, at long or too-long range, with most characters I feel like my Kyo is at a disadvantage. His pokes aren’t great range or amazing priority, really, I’ve never found his dash all that useful and his roll (I’m in C-groove here) is cack. Like I said, I’m trying to switch to N-groove. That’d help. I’m sure his little jump D.Fp would be a lovely move. Or just even the ability to small jump, and especially run close.
Anyway, it’s at this time I’d get screwed up. I’ll then spend my time working a way back in. Kyo has a very nice magic elbow in J.Jp - beats most things in the air (even Benimaru’s magic knee) and stays out for a long time. Also I quite like his flying RH, as long as you’re above your enemy this is a nice kick. But usually I get smacked around at this point - I need run, I suppose. Also, since my opponents sense this in my play too, once I manage to get close they just do everything they can to reset the fight. And that’s not toooo tricky, usually. What I really need is to tighten up my pressure game and try to close the holes which allow people to escape to greater range. Maybe his kick specials would be cool here since I could hopefully stop their jumping away so easily… is that what people would usually do?
OK, so, main thing is for me - get in close. Once you’re in close - try to stay there as long as possible. While you’re still there - maintain a nice flurry of little pokes, flaming punches, occasional cheeky throw, and the odd super to counter. If they jump away - try to dash back in close.
I’ve been doing the j.Rh, s.Rh xx qcf.Rh, Rh, Dp.Rh combo - superb damage. You can try cancelling your poking s.Rh all the time in the hope of catching the double flying kick - and pretty much anything hits afterwards, even standing Fp. But, if you’re blocked then prepare for some punishment.