Law / Polish resident card questions (got accepted at Jagiellonian University) [5]

@swissi: A student can get resident card (1-year period). After 1 year you have to apply for extension.

Law / Apply Karta Pobytu (Residence Card) for my wife [7]

Yes. There's a paper confirming that a person can entirely apply and receive resident card on behalf of another person. You will need your wife's passport number and her signature. You can ask for that paper in Immigration office (same place that your wife applied for the card). Good luck!