Are air batlles possible?

Yes.

Yes. And with Magic

Yes, they are possible and I just did so today. Legendary Dragon vs Legendary Dragon, I was riding on one and fast traveled to a bandit camp when another spawned. My dragon and the other one joined up to wipe out the bandits and then I attacked the other dragon. It was pretty darn epic, one of the more epic fights possible in Skyrim. Even flew around Dragonsreach itself for awhile, before the other Dragon was forced to land and eventually killed just outside of Whiterun.

I got bored of it rather quickly tbh. Very repetitive, limited to zero control and the dragons don't exactly.... have an epic fight. I've seen pigeons do better

I haven't had more than one Dragon vs Dragon battle yet so that was quite epic for me. The way my dragons behaved wasn't too bad though, although for most of the time the enemy dragon wasn't attacking...but yeah, I can see it getting boring rather quickly, especially if you aren't a mage since you'll have nothing to do while the dragons duke it out.

Ah, one battle would explain why it was epic I've had four or five now. I wanted to test out the capabilities of dragon riding so I made the dragon fast-travel to a Word Wall. Of course, a dragon then spawned and POW! Battle commenced... kinda. Lucky for me, the character I was using at the time was a mage. Other characters will, as you say, be bored poopless when engaged in battles. I know you can use shouts, but the best ones are locked out during dragon riding

Is it possible to use Storm Call/Call Dragon during dragon riding? I've only tried Unrelenting Force and Drain Vitality...kinda dumb of me, since they missed the target hopelessly.

Also, instead of Dragon Riding, the Bend Will Shout could be used to make Dragons become sort of temporary followers, which might be less boring. I was out dive-fishing near Whiterun when a Dragon attacked me and I was in my Alchemy gear, unprepared for battle, so I used Bend Will on it instead. It began following me (awhile after I didn't ride it) and toasting all the mudcrabs and slaughterfishes that tried to attack me as I swam along the river. Pretty amusing, but also the first time I tried such a thing. The other times I was just fighting bandits/forsworn with Dragons and yea it was kind of boring but also amusing in a way...the dragons seem so much more powerful when I'm riding on them compared to when I'm fighting them.

I have been told you can use Storm Call when Dragon Riding but alas I haven't tried. I will look.

On the few occasions I used Bend Will on a dragon to make it a temp follow, it caused more then problems then it solved. I prefer to use BW on a Giant and then attack a mammoth. T'was good fun

Alright, perhaps I will try it out myself some time.

Haven't tried Bend Will on a Giant yet, should be fun! Could use it to set people up for a Shout-off-Cliff as well. Assuming it can be made to cool down before the effect wears off.

Bend Will calls down quickly when not used on a dragon. Even then it cools down enough for you to use a few shouts. Giants, mammoths, most NPCs, vampires etc. Yet to find something I can't bend the will of. Well, except Alduin and his spawned named dragons.

Nice, there could be a few new ways to have fun with this shout then. What about Durnehviir? Would be cool if we could ride him around the Soul Cairn.

I have been told no, but will try it anyways. Could be interesting if you could! As you say there are quite a few new ways to have fun with BW