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#1.
A --------> B <------|------> C, we all know = A<------|------> C, but what about ~A --------> B <------|------> C, does this = ~A<------|------> C? or does it = something else? Please explain. Thanks.

#2.
~A------>B
CONTRAPOSTIVE = ~B------>A

~B------>C

Can I link them to be A------>~B------>C? Please explain. Thanks.

#3.
Also, what about this A------>~B------>C? Does = A------>C or A------>~C? Please explain. Thanks.

Let's look at those letters from a different angle.

Suppose A = human, B = mammal, and C = amphibian.
From A <-|-> C, you're right; no humans are amphibians.
Not sure where you're getting ~A from because you can't just negate one side out of the blue,
though ~A <-|-> C works in the same manner as A <-|-> C, except you'd have to tweak the definition.
If A = animal and C = dog, great; something that is not an animal can never be a dog.

One way to make double arrows easier is just to break them into two arrows.
~A -> B <-|-> C can simply be written ~A -> B -> ~C and ~A -> ~B -> C; you almost got it.
With this, take the contrapositives to get C -> ~B -> A and ~C -> B -> A.

The last one is A -> C.
If A = gold, B = NOT ON the period table, and C = element,
then gold (A) is ON the periodic table (~B), and if it is ON the periodic table, it must be an element.
Therefore, it makes sense to say that gold IS an element, not the reverse.