This has nothing to do with Catholic "mysticism" at all, but with an ecclesiastical procedure that allows Catholics to remember this man yearly, to ask his intercession, and to praise and emulate his example. But if you want to know more about Catholic mysticism proper, you can start here.
Is JP in purgatory now? If he is, and he is made a saint, does that usher him into heaven immediately?
A beatification is a declaration that a person is among the just. Those in Purgatory are among the just so in principle, it makes no difference if he has been admitted to the Beatific Vision ("heaven") or not. However, it can be argued that we pray for a soul in purgatory, and ask the prayerful intercession of someone who has been beatified or canonized, so the implication is that the person in question is in fact, beholding God in heaven.
The declaration of canonization or beatification doesn't "promote" the person in heaven, but in a sense, promotes the person "on earth," in the eyes of the faithful. There are thousands of saints in heaven, most of whom have not been canonized nor do they need to be in order to enjoy the Lord's presence.
When exactly does that happen?
We can't directly know until "after the fact." The proof lies on the miracle brought about by the candidate's intercession.
Can departed Catholics in purgatory hear your prayers?
Good question. My personal, private opinion is a qualified "yes", provided that we pray *for* them, not *to* them.
I have been a non-Catholic all my life and Im sure the answers will generate more questions. I do not understand how Catholics think about these things at all ...
I'll attempt to answer all the questions you may have, or send you to those who know more when I can't.
Yours in Christ,
-Theo
Ah ... I see. Thanks for the clarification, that removes lots of assumptions on my part.