Proving $[0,2]\big/[1,2]$ is homeomorphic to $[0,1]$

Here is a simplier argument using a classical lemma.

You have shown that $f$ is a continuous bijection. Then, we now that $[0,2]/[1,2]$ is compact, as it is a quotient of a compact space. Also, $[0,1]$ is Hausdorff, so any continuous bijection $[0,2]/[1,2]\to[0,1]$ is a homeomorphism.

You can find here a proof of the lemma (it is easy).