Is it permissible to work with a medical degree that I obtained by means of bribery, and does passing the evaluative exam make it permissible for me to work with it?

Bribery is forbidden and is only permissible for restoring a right or averting harm. Your payment of a bribe to obtain a certificate without effort necessitates repentance. As for working with that certificate, it is permissible if you are qualified and capable of performing the work in the relevant field; otherwise, it is not. The field of medicine is dangerous and sensitive, requiring sufficient expertise to avoid harming patients.