Have the highest autoattack range?
has the highest at level 1, 
has the highest at level 18.
, 
, and 
can all spend mana to surpass Tristana's late game AA range.
Have 350+ move speed without any masteries/runes/items?
and 
are the only 2 toons with a base 355 MS, the highest.
If you add in abilities - plenty of champions come with a %MS steroid in their kit somewhere, and gap-closers can be translated into a %MS bonus if you factor in their casting time and distance. I'd say a solid ~80% of the champions in LoL come with a form of mobility somewhere if you're not ruling out abilities completely, and as the lowest base MS is only 325 - getting past 350 is going to happen with any skill that grants an 8% MS bonus or greater.
Gain a stat from another in the manner that Ryze does?
, 
, and 
have permanent stat conversions in their kit.
A solid ~20% of the champions in LoL have 1 skill with really unusual scalings as well. HP, Armor, and MR are used for skill power in champions like Leona, Vi, Malphite, Shen, Olaf, Volibear, and a few others. Yasou gets CDR from AS, Riven and Diana get an HP shield from offensive stats. A few champions also do physical damage with AP scalings or magical damage with AD scalings.
Have abilities or passives that increase a stat and how?
Almost every champion has a steroid in their kit somewhere.
Mages are the only archetype in the game that doesn't come with an auto-attack steroid more than 50% of the time, and with stat bonuses you've got more than just stuff that enhances auto-attacks.
Defensive steroids arn't uncommon either - it took me ~6 seconds to find a tank, fighter, assassin, marksman, mage, and support with an armor/mr steroid in Leona, Jax, Akali, Graves, Annie, and Sona. There's plenty more.
Can perform a win "rush"? (meaning early/mid game its vastly more probable to win)
Champions are pretty evenly divided in regards to their power spikes - include early/mid game and you're looking at a solid 50% of the champions in the game.
Generally speaking bruisers and assassins are mid-game carries, mages and marksman tend to be more late-game carries.
and 
are the 2 champions capable of pushing down an enemy Nexus the fastest. In a 1v0 game, they can both do that in under 6 minutes.
I say that because Riot hosted a thread asking people to test that out, and that's what the results were with ~100+ participants.
Act as a hypercarry and at what point of the game?
Hypercarry tends to imply a champion your team has to carry to the late game to matter, but then can carry your team once they get there.
Another way of stating that is they're useless without items, but scale with items better than any other champion.
It's not a matter of what point - hypercarries are late-game monsters 100% of the time.
You can realistically carry your team with any roll, hyper-carries are a definition you use for a specific time when that carrying happens.
Trying to plan on which champs to "main"- so far I do Jinx because she ADC (top/bot/jungle)and I have gotten so good at Kill-shots with my rocket that I can't NOT have fun. I realized that when I lane, I do so very aggressively. I also really do focus on stats and passives in a build more than actives even though ,yes I know, actives have incredible effects (DFG, zhonya, gunblade, etc).
Well - the general rule with passive bonuses is you get demolished by anyone using a lot of active stuff, and only really beat them if you can stick to them like glue long enough to overcome their burst, or you have someone like Nautilus playing body-guard so you can safely pew-pew behind them.
As for learning to use actives, Dominion really helped me with that.
Best champion for passive power is 
though. 300 AD steroid, 40 armor/mr shred, 600 HP, infinite scaling Q.
.
