Is the money received by the person I work with from farmers permissible or forbidden, and is the extra money I take from him forbidden, given that my work leaves me with a very narrow window for prayer?

It appears that the doctor pays the man a wage on the basis that he provides a free service to the farmers, and this may have been stipulated. If this is the case, then the man is not permitted to take anything from the farmers except with the doctor's permission, because he is a special employee of the doctor, and acts as his agent. He should not accept gifts for his work except with the knowledge of the employer, due to the Prophet (peace and blessings be upon him) saying: "Gifts to officials are ill-gotten gains." As for delaying prayer beyond its time for the sake of work, this is not an act of worship, and God's obligatory duty must be prioritized.