The “à” is definitely necessary in the sentence because it is the whole point of it. Read it like:

Parce que la préposition « à » indique un mouvement

In the second sentence, faire à is indeed to be done:

Les devoirs sont à faire à la maison : Assignments are to be done at home (i.e. they are homework).

De is required in the third sentence as it is written.

La forme la plus correcte est de lire à la page cinq

in the first sentence, you must consider "préposition à" like a noun. Personnally I would write "à" in italic in this case to show that it is not used as a preposition. Indeed, not all preposition indicate a movement: this is a specificity of "à".

Parce que la préposition "à" indique un mouvement.
Parce que le "à" indique un mouvement.

Because "to" indicates a movement. (I apologize if my English is not correct)

for the second sentence, your interpretation is correct: ‘à faire = to be done’

for the third sentence, you could also write

La forme la plus correcte est: "lire à la page cinq"

but otherwise you need the "de" like in :

Il est correct de faire…
Il est important de faire…
Il est mal de faire

etc