Does IP price have anything to do with their champion viability? ( I.E. : Ashe VS Jinx, even though I am saving for Jinx anyway because that cinematic o.O) (Also, Heimerdinger and Zyra...that confuses me how they have near identical Q/W/E/R but Heim has it way better....)
In a convoluted way, yes. Very difficult champions tend to have better rewards for being played well to make up for how awful they are when misused. Champs that are very easy to use, by contrast, tend to be mediocre regardless of the player (after all, a champ that could consistently carry regardless of their player's skill would be pretty awful for the game). So, because easier champions tend to be cheaper and more difficult champions tend to be more expensive, the correlation exists.
Why do I see people who play pure damage have 21/0/9 or 21/9/0? (I see a total of about 17 points I would want to put into the damage tree for my damage characters)
You have to invest 21 points in a tree to get its 'capstone,' which are some of the highest impact single points you can invest. 3% increased damage across the board, 15% tenacity or mini-mobility boots for just one point is pretty insane. There are a few champions that can get away with stuff like 0/14/16 (most of them supports), the vast majority of champions are best going 21 in one tree and 9 in another.
Oh, big one for me! How do I identify what champions hard counter others? (Like Yasuo dominates projectile based Champions(Ahri was my main until I lost 5/6 times vs Yasuo) which hurts a bit)
Obligatory - every matchup is winnable with jungle intervention. Theoretically.
Practically, a lot of this stuff is situational. Some champions have very polarized matchups - Teemo dominates melee auto attackers, Kayle beats ult-reliant assassins, etcetera. For a lot of champs, however, it's dependent on your teammates and the skill of you and your opponent. Thresh is a counterpick to Zyra if you have an ADC with significant burst damage, for instance, but he's not actually gonna do that much if your ADC is Ashe or somesuch. Elise into Amumu is a good matchup if you have the stronger laners, but it's not really that great if your lanes are permanently pushed in. Anything into Lee Sin is an excellent matchup if you're playing against someone in Bronze. And so forth.
