Sorry, I had to split my answer in two parts, because it exceeds the length limit.
> Perché non si può dire: "le faccio farsi la doccia" o "faccio che si vesta"?
In theory, you can say "faccio che si vesta".
In the first person, though, it sounds very unusual (I never heard or read it). On the contrary, "faccio in modo che si vesta" sounds familiar.
It's just a matter of usage, maybe due to the huge emphasis typical of "fai che…", which could have inhibited the usage of the phrase in the first person, where it can't have such emphasis.
If you search for examples of "faccio che" + infinitive, beware of the regional Piedmontese usage of this construction with a different meaning, which isn't part of standard Italian.
As regards "*le faccio farsi la doccia", it seems to me that there is a problem with it, because it offers no acceptable positioning for the reflexive pronoun:
a) on the one hand, causative verbs always attract all clitics and in "*le faccio farsi la doccia" 'faccio' would fail to do so.
b) on the other hand, in your sentence, if the reflexive pronoun climbed ("*le faccio farsi" -> "*le si faccio fare"), it would have to change to its non-reflexive counterpart ("*le si faccio fare" -> "*le le faccio fare"), because the subjects of the causative verb and of the infinitive aren't the same, and this would produce the clitic sequence "le le", which isn't part of Italian.