Please next time, tell us that A,B,C are events, and not random variables... And the notation of where A is an event is not very common. So you'd better write P(A) instead. If A,B,C are indeed random variables, then your second question is not valid.
What does "A|B|C" ? Which event are you conditioning to which one ?
if A and C are independent (that can be proved by using Bayes's formula)
Use the law of total probability (check that up on wikipedia)
If 2 rv's are uncorrelated, then yes their covariance is 0....... read the definition of a covariance or a correlation (one is equal to the other multiplied by some standard deviations)
If 2 rv's are uncorrelated, it doesn't mean that they're independent. If you're looking for some formulas, you can just check on the internet, they may be plenty of them.
With a formula, it gives this :
. So if then , and this doesn't imply independence. But independence implies this.
If you want a more word-ish explanation, it's good to know that there are several kinds of correlations, the most common one being Spearman's correlation coefficient (there's also Kendall's, for example). This correlation "measures" the linear dependence between 2 rv's, but they can be dependent in another way.
*hand-made explanation* : You can imagine two points in a plan. It's possible to link them by a straight line, but you can also link them by some curves. This is the same kind for dependence : you can have 2 rv's that are not dependent "on a line", but that are dependent "on a curve".
Note that in a general way, independence between two rv's is defined as follows :
Meditate all of these.