Can we prove "If $A = B$ then $C=D$" by assuming $C=D$ and showing $A=B$?

No, what you are describing is the logical fallacy of affirming the consequent.

If you have $P\implies Q$, that doesn't mean that $P$ is true just because $Q$ is. Example in propositional logic

It rains, therefore the road is wet.

This is perfectly valid, however just because you see the road wet doesn't mean it has rained. After all I can grab my hose and spray it all over the road for an hour without it having rained or be raining.

With your "route" we prove that 1=0 as follows:

1=0, hence 0=1.

We have:

1=0

0=1

Adding these equations, we get 1=1, which is true ....

FRED

No.

Let us take an example -

If $x = 0$, then $\sin x = 0$. But if $\sin x = 0$, then $x$ is not necessarily 0.

If we start by your method to prove $\sin x = 0 \implies x = 0$, we would end up proving it despite it being a false statement.