Is the card issued to the company with the employee as an authorized user or does the employee have to qualify for the card?
The company qualifies for the card. Individual credit scores are not checked. The company can request the card to be cancelled at any point in time. The obligation of payment is with the company. This is the reason that card issues banks are not worried about scores of individuals. From a risk point of view corporate cards are less at risk of default.
If an employee declared/declares bankruptcy and part of his debt was to the bank that issues the cards for his company, is he still able to obtain or keep the company card?
In theory yes. As explained the individual's status has not direct bearing on it.
In reality, most organization need an employee to declare significant events. So if you inform your company [or they get to know]; the company may request card to be cancelled. This would be in line with most company policies that the chances are high that an individual may use the corporate card for personal purposes.
Most companies have a policy to strictly not allowing personal use of company card. The reason being if you swipe a amount for personal transaction on company card, the company is liable to pay to card issuing bank even if you don't pay the company.