All four of your examples are correct and proper English.
The clause can be omitted only if it's 'someone/something is' but not 'someone/something does'.
This sounds like a pedant's rule, a prescriptive statement that represents someones opinion about how the language should work. However, this doesn't reflect the way that English actually does work. Native speakers of English happily produce and understand as clauses in which a verb other than be is omitted. The actual requirement is that the verb can be omitted so long as it's the same as the verb in the main clause:
I sing the same song as you. [sing]
The girls take the same classes as the boys. [take]
But in the following example, the verb cannot be omitted:
We sing the same tune as the pianist plays.