Ah, fellow RPG lover, as I can see. I totally understand you, brother. 
The lack of these numbers annoys me too.
I will start with a short version of the answer to your questions. Please be patient to my English, it is still in development :)
About the champions skill and how they scale in terms of damage. There are hidden numbers behind each skill called modifier that determine the output damage of each skill. What we know about this modifier is that it varies between different champions, different skills, and even between different hits in one skill. In most cases it is initially lower as you develop your champion, and when your champion hit 6/60 his modifiers (just like some of his base stats) increase to some extent.
I'll give you an example. That lady Elhain which you chose in the beginning because she is an beautiful elven maid with bow (you chose her, right 
) has three skills. Her second skill (we call them A1, A2, etc.), so her A2 shoots one arrow first, then shoots arrows at all targets. He have noticed that the first arrow does more damage than the each arrow from next multi hit. But why, it's same skill, just different hit? This is where this hidden multiplayer comes into play. The multiplier behind first shot is 6.3 when not fully leveled and 7.7 when she is fully leveled 6/60 and for second multi hit is 3.5 and 4.4 respectively.
For every champion and every skill out there, no matter how they scale (what stats is mentioned in the description: ATK, DEF, HP or combination from those) the damage is calculate like this: "respective skill * modifier = damage output". This is simplest explanation possible and there are additional variables, but they are of less importance, with less impact to final damage and for some of them i don't have enough understanding to explain them.
The practical application of the above: the damage scales linearly and at some point it is more beneficial to focus on crit. chance and crit. damage. Most of the end game players which advise I follow recommends following path: let say your champion scales with ATK. Make your additional atk two time champions base atk, then go for crit. chance and crit dmg, building them more or less simultaneously. When you hit 100% crit. chance and around 150% crit dmg focus on building those atk and crit. dmg more or less at the same time with a relatively equal increase. If you increase your attack with additional 20%, then try to increase your crit. dmg with at least 10% more. This is based mostly on some users tests and as many other things in the game, where we don't have official numbers and explanation, is more or less just unofficial and recommended, not something you should follow at all cost. And obviously, it is just explanation of how in theory you can get most steady increase in damage. In real game this is hard to follow "by the book" as English speaking people tend to say, i think.
As for your other question. We don't use "dodge" as explanation here, you rpg maniac hehehe, no offense!
I think its better to give you the official video of how accuracy and resistance work, it will be lest painful than reading my explanation :)
*edit - ah, i need to improve my typing skill, @Ethan is faster than me :)
