Can I find $|A \cap B \cap C|$ if I know $|A|$, $|B|$, $|C|$, $|A \cap B|$, $|A \cap C|$, and $|B \cap C|$?

No, you cannot.

Let, $A=\{1,2,4\}, B=\{2,3,4\}, C=\{3,1,4\}$ so that $|A|=|B|=|C|=3$, $|A\cap B|=|B\cap C|=|C\cap A|=2$ and $|A\cap B\cap C|=1$

Now, let, $A=\{1,2,3\}, B=\{1,2,4\}, C=\{1,2,5\}$ so that $|A|=|B|=|C|=3$, $|A\cap B|=|B\cap C|=|C\cap A|=2$ and $|A\cap B\cap C|=2$

No, you cannot. Let me give you a counterexample. Suppose $|A|=|B|=|C|=3$, $|A\cap B|=|B\cap C|=1$ and $|A\cap C|=2$. Then there are a bunch of possibilities that meet these conditions, but we are only interested in two of them. Let $A=\{1,2,3\}$, $C=\{1,2,4\}$. Then both $B=\{1,5,6 \}$ and $B=\{3,4,5 \}$ meet the initial conditions. With the first $B$, we have that $|A\cap B\cap C|=1$ and with the second one $|A\cap B\cap C|=0$.