What is the ruling on money paid by an employee as compensation for their negligence at work? Is it permissible to give it to the needy instead of returning it to the government? Is it permissible to give it to the poor in other countries? And is the official's condonation of it sufficient?

The basic principle is that one's financial liability to the hospital is absolved. However, if you are certain or highly assume that the hospital has a financial right over you, you must pay it or seek absolution from an authorized person. There is no need to mention the details of the right, and a Muslim is encouraged to conceal their own affairs. Charity does not suffice for returning rights to their owners if you are capable. If you are unable and despair of doing so, then give it as charity to the poor, and they do not necessarily have to be from the poor of the same country.