MEW 1.0.0

Don't I remember hearing that this particular Mew can be shiny?

AFAIK It probably can't.

The PID generation method and RNG conditions of My Pokémon Ranch are at-time-of-writing unknown. Like all other Hayley trades, this is likely to be shiny-locked, and no one has ever come forward with proof of receiving a shiny Hayley Mew or Phione, but a shiny-lock has never been confirmed for sure.

Edit: Kurt, Sabresite, and Admiral-Fish discovered Ranch uses antishiny ARNG, which is the same method that causes Gen 4 Mystery Gift anti-shininess.
https://bulbapedia.bulbagarden.net/wiki/Pseudorandom_number_generation_in_Pokémon#Alternative_pseudorandom_number_generator_.28ARNG.29

Wherever I look for the PKM Ranch Mew I only find one for "Diamond", even though I transferred it to Pearl. Every time I change the "Found location" to "Pearl" it says that the move hypnosis and teleport are illegal. Why is that and how can I fix that?

7 hours ago, Ape said:

Wherever I look for the PKM Ranch Mew I only find one for "Diamond", even though I transferred it to Pearl. Every time I change the "Found location" to "Pearl" it says that the move hypnosis and teleport are illegal. Why is that and how can I fix that?

The origin game is always set to Diamond even if you received it on Pearl.