Golfer AbbaS, I have heard people ask whether pushing notes into the crevices might be assur, not because you’re undermining stability of the wall, but because you may inadvertently approach an area that has Kedusha. I don’t know the answer; I’d be curious if anyone else has heard anything like that.
The amount of damage is minuscule,which is why it is permitted, but over time with thousand of people doing it the crevices are enlarged. As far as sticking their hands deeply in the crevice, I have not done it nor seen anyone do it, but I assume the just put it on the edge of the crevice and the next person does the same pushing the first one deeper into the crevice. I assume the notes dates from the times concept of writing when they use to write kvittles to the Rebbi. If it was good enough for the rabbi how much more so for Hashem.
As far as the wall itself it surrounded the courtyard and any expansion required the approval of the Bais Din of 71 and was paid out of the excess funds from the half Shekel poll tax and should have the same kiddusha as the Bais Ha”Mikdash.