-Can be played in two or more roles -Hard to master and learn -Not a tank (Can be a top lane fighter, etc.) -Can go balls to the wall and just jump in the enemy team once you know how to use them -Isn't picked or banned super often do you can play them more (this isn't a real requirement, just a nice add-on) -Isn't completely useless if behind
Thanks in advanced
-Can be played in two or more roles -Hard to master and learn -Not a tank (Can be a top lane fighter, etc.) -Isn't picked or banned super often do you can play them more (this isn't a real requirement, just a nice add-on) -Can go balls to the wall and just jump in the enemy team once you know how to use them -Isn't completely useless if behind
These 4 are the closest champs to fit all those criteria. A bit of a tough fit though, especially with 'jumping on the enemy team' often meaning a champion is either a tank or useless when behind.
But all 4 have fairly versatile kits that either cover the strengths of a lot of classes or can be played in multiple lanes, sometimes both. All 4 have a lot of room for mastery. Everyone save Nidalee has a solid foundation to pick up and play them OK right away too. None of them are tanks. Some are popular, but they're rarely banned. Ezreal and Lee Sin usually reach a ~40% popularity rate at some time in every season though. All 4 have jumping capabilities and can go deep. All 4 tend to have enough poke and/or utilities to be an asset to their team even when behind.
is also worth testing out whenever you want to try a new champ, or just discover what you like in a champ. She's the definitive jack of all trades in the game (all with the same build), can pull off more lane positions properly than any other champion in the game, and covers more champion archetype strengths than anyone else.
She doesn't have any hops so she can't jump into the fray in that sense, but she covers everything else you listed.
Sion.
Hecarim? -Top Lane and Jungle -Learning when and how to initiate takes some learning and if you cock it up then you risk getting your team killed. -The BEST initiator and can run roughshod over the enemy team, also highly mobile and fast as hell. -People seem to forget about 'ol Hec in favor of like Veigar and Teemo and the like. -Hec's mobility affords him map coverage only rivaled by, I think, Rek'Sai and her tunnels. Enemy team engaging on bottom lane? Run top and push the turrets. Enemy comes up that way? Run through the jungle and Ult over walls to escape.
Just my opinion though.
is a jungle/top/(maybe mid) champ, who can be built tank or damage or bruiser. He does have the balls to the walls aspect, but he is a little useless if he builds damage and is behind. If you are worried about getting behind, you should build stuff like dead mans and titanic hydra. Stuff that give damage, but also give health/tank.
I actually would say Karma fits in this. Not exactly a go balls to the wall and jump in champion, but she can certainly do a lot with a lead and can survive being in the fray quite well.
Twitch also - Twitch jungle is pretty solid right now. He can also be played ADC. While I don't know about jump in the enemy TEAM, but definitely an enemy or two. He doesn't fall off late game, but his early game is on the weaker side.
Rumble - he can go mid in certain match ups though most often is played top. If he falls behind his ultimate does a ton of damage.
Annie - Low skill cap but she fits a lot of the criteria.
Top/Jungle
Jungle and Top
Tho top only
Top/Mid (Tho it's all written in rules - if you pick Riven you're camped as hard as possible)
ADC/Mid
Top/Mid
Top/Mid/Jungle (not exactly "hard to master" but isn't banned like hell and is strong so...)
[ 
Top/Mid and even jungle - not hard to play tho and I guess mostly banned]
Top/Mid/Supp - Be rock, feel rock, rock solid. You can go full AP and just go rock hard on their asses. Tho not hard. But not picked that much I think
Top/jungle - (His ult is essence of league of legends)
Top/Mid
Top/Mid/Jungle, isn't exactly easy, and I don't think he's banned as much
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Tho banned and picked a lot cause he's strong. Can go jungle, can go top]
Not sure about his ban ratio atm in NA, but y'know you can go Top/Mid and such.
Mid, top, even seen her in jungle, so...
ADC/Jungle, strong af, especially in lateNot every single one of those is hard but still, if you need champion for whatever role, here they are.
You can always pick 
and just go, no one knows how to play Yorick anyway...
: Top/Jg. Builds AD/Attack speed. He's not the easiest to learn. If fed he can 1v5 and win. If behind he can split push like a mofo and take turrets (Once took 2 turrets in the time a team fight was happening)
: Top/Jg/(supp?). Build AD/Attack Speed. Timing his ult is harder than it seems (many trynd's are bad cuz they never get the most out of their ult). He can 1v5 if fed, split push or just straight up murder carries (the later takes positioning and skill).
is limited to two roles and has an existing pickrate.
is quite useless when behind, but hey you can always press R.
I would say nidalee (I don't play her at all).
She can play in any lane best as a jg but decent at mid or top. Has a high skill cap. Can dive into the enemy and kill them. Not a big ban rate unless your in lcs. She can suffer from behind but if you are behind just stay back and launch spears at max range and watch your enemies get one shot regardless.
This is the only champ I can think of that truly meets your qualifications. All of the other champs I see on this post are pretty situational. Nid on the other hand is pretty much blind pick worthy becuse her kit is so versatile.
irelia.
