Learn Russian: Я схватил было ключи, но вспомнил про телефон. - I had grabbed the keys, but remembered about the phone.

Here, было is not a verb but a particle of thwarted action. When placed after a past perfective verb like схватил, it shows that an action was completed or just about to be completed, but its expected result was immediately canceled or interrupted by the next event.

Since the action was interrupted, shouldn't we use an imperfective verb instead of the perfective схватил?

No, this specific construction almost exclusively uses past perfective verbs. The perfective aspect shows that the initial action achieved its result (you actually grabbed the keys), but the particle было acts as a rewind button because the expected follow-up (leaving the house) was interrupted by remembering the phone.

Could I put было somewhere else in the sentence, like before the verb or after the keys?

It is most natural right after the verb it modifies (схватил было). You might occasionally see it before the verb or after the object in older literature, but directly after the verb is the standard and most natural placement in modern Russian.

What is the difference between saying вспомнил про телефон and вспомнил о телефоне?

They both mean the exact same thing here. про followed by the accusative case is very common in everyday conversational Russian. о followed by the prepositional case is slightly more neutral or formal, but they are fully interchangeable in this context.