Will a credit card company offer a card if a person has multiple debts?

The credit card company will assess the person's financial ability to know whether or not the person can afford it. Or, the person will get a card with higher interest rate.

The credit card company can offer a card even if a person has mortgage and an auto loan. Credit card company will check if the person makes monthly payments on time.

Yes, if that person pays the bill on time, the credit card company will offer. basically what they check is the person's creditworthiness before offering the credit card.

The credit card company can surely provide a credit card to a person who is already in debt (mortgage, car loan, or personal, etc.)
But they will check if the person is capable of making payments on time for each debts.