Wikipedia is clearly not a better idea, unless you want yourself big spoilers for 10 seasons of Kamen Rider shows in a row, so it wont be interesting to watch them afterwards. In info I posted, I kept spoilers at minimum.
Jumping in game was done just for its being there, it wasnt done with usual concept where jumping is essential part of evasion and anti-air game, all that hopping over projectiles, empty jump-ins mindgames etc, its not about this game. But on other hand, jumping here is far more advanced than in MKDA-MKD-MKA games, even though those games went “why jump when you can sidestep attacks” too.
As for air attacks, most of them are somewhat basic, breakdown is like this:
-close-ranged air attack, can start combos if done closer to ground
-vertical falling air attack, same
-rider kick (flying kick that moves diagonally down) - can be used to quickly get to distant opponent, plus can be used as runaway tool if you air recovered during opponent’s juggle combo. It homes on opponent’s location unless its not too far
-air projectile; homes on opponent’s location unless its not too close or too far
-something unique that most characters dont have (like Ryuki Survive mid-air dragon summon, Hibiki’s air flame breath that slows falling speed, etc)
To me, probably most important aspect of air attacks in game is the fact that you can do them after air recovery, and you can air recovery during opponent’s juggle combo if you are lifted enough. So basically, falling air attack gives you ability to counterattack opponent in this moment, at risk that you can be yourself attacked (Blade King and Ixa Burst have some strong anti-airs as Support Attacks), and dive rider kick-type air attack gives you means to runaway in air after air recovery
As for shows,
you get 2000 season here (this isnt hardsubbed, and subs that are present need some grammar correction, I did some notes about what means what in those subs while I watched it, can send them if you need)
you get 2001 season here (hardsubbed)
http://henshinjustice.com/forums/showthread.php?t=27549
you get 2002-onward seasons here (hardsubbed)
http://tvnihon.com/
Most shows consist from TV series season, a movie and a TV special