Is using the first card, which was obtained through bribery, unlawful (haram)? What are the consequences of using it for studies and other purposes? Is the second card, for which no bribe was paid, lawful (halal), or is there no objection to using it?

There is no objection to benefiting from the card. What your grandfather did by paying a bribe to expedite the transaction without your permission does not incur sin upon you. He must seek forgiveness and resolve not to return to such an act, for bribery is one of the major sins. There is no sin in asking for a waiver of turn from those waiting. It is permissible to benefit from the card in lawful ways. You must cease obsessive thoughts and pay no attention to them.