Noob questions from a non-noob

So I've been watching baseball for quite a while, and I would say that I know a fair amount about the game. So it astonishes me to realize that there are some very basic questions to which I just don't know the answer. So, here's a couple of noobish questions for anyone to help me out with:

1) I understand that a 2-0 count is good for the hitter, because the hitter is ahead and therefore the pitcher is under more pressure to throw strikes, i.e. pitches that are more hittable. What I don't understand is why a 2-0 count seems, colloquially and in media, to be a better count for hitting than a 3-0 count. It would seem to me that the dynamic would only continue to progress in the direction of the hitter's favor, yet my anecdotally-supported perception is that this is not the case.

2) Similarly, why are certain counts better to steal a base on? Is it indirectly related, due to the pitcher have more focus on throwing strikes to the batter?

3) Why, specifically, is it a good thing to "establish the fastball" early on in a pitcher's start? The O's seem specifically to have a strong organizational philosophy of doing this, and often our starters throw all or nearly all pitches in the first inning as fastballs. If pitching is about keeping batters off-balance, why start with the ultimate balanced approach?

1. 2-0 you still might could get on top. 3-0 you don't have much chance.

2. If the pitcher is ahead, 1-0, 2-1, even 2-2... he can throw his breaking ball. Which is slower which is when you want to steal.

3. The best thinking... is get the guys out the first time thru the lineup with just fastballs. Second time thru mix in your slider/curve/change. Third time mix in everything. Whatever you have.

Here's a question:

When a pitch is thrown and the ball barely scuffs the dirt, the catcher asks for a new ball. If a pitch is hit on the ground to an infielder and the batter is thrown out at first, the ball is thrown around the infield and given back to the pitcher and he keeps it to continue pitching with. In both cases, the ball hit the dirt, yet only a new ball is requested in the first instance, why?