What do you mean by "that it has on itself"? When I open the Palico Skill screen I see it being divided into three part, and the last part has his forte skills. So the Sub Palico technically won't use those and only the top quest skills only?

I mean that the sub palico does not have access to the skills that the Ace palico has. If the Ace has Demon Horn but the sub does not have Demon Horn then the sub cannot use demon horn. The sub can use all skills that are listed on its stat screens.

So I'm right by saying that all level 10 Fighting Palicos I'll recruit in game will have the same stats on that level?

All level 10 fighting forte palicoes will have the same stats, regardless of how you got them.

But I'm confused by your statement - according to the guide if I have 4 small red icons they should be worth 1 Extra large?

Yes, though that is a very inefficient way of doing it. If you have access to large or extra large then there is no reason to use smal.

I'm only at the very start of the game and have only 2 routes to pick from. But I never encountered a route with a single color. Why this method is better then trying to pick all colors associated with the Route? It means otherwise I'll have to lose one fight?

As you get further you'll see more routes that have a single weakness for most, of not all, monsters. Some monsters might have two colors, but the idea is to find a route where 2 or 3 monsters are all weak to one color, making it easier to defeat all of them.

1. The Leadership skills I wasn't fully aware of but I just checked. They're exclusive to the Ace and he will always have them, even if you change his forte.
2. Manual seems to have higher success rate.
3. I've found Paralzis Edge to be quite useful. Sleep Edge and Blast Edge might be good too. Last Stand should usually be avoided. I'm not too sure past that.
4. For quests, melee/ranged is based on forte, I believe. Boomerangs do less damage per hit, but should be able to stay alive better and have better accuracy.