During the campaign you'll sometimes see a short sort of 'cutscene' of a given planeswalker, followed up by a 3 page biography that presists until you press "Continue". On the first page of this bio, there is always a "Planeswalker" card, which matches whichever planeswalker bio you are looking at, and has some numbers and abilities on it that don't make sense to me. I've played a fair bit of the campaign, and not come across a mode where these cards are available. Is there a mode in the game where these cards are available and used?
No, they're not. Wizards of the Coast has the full decklist on their site, and there are no planeswalkers in any of them.
It's pretty confusing as to why they show them in the character intros: my guess is that they're intended to be minor advertisements for the physical card game or Magic Online.