It is permissible in Islam to become a doctor.
In general, Islam never told that a believer should abstain from studying medecine, and there are many Muslim physicians worldwide; rulings only concern some points as those you have mentioned in your question:
- You may draw and study drawings of any parts of the human body; drawings of the entire body may be acceptable if it is essential to draw the entire body for medical purposes but it should be avoided
- It is disputed to what extent working with a dead body is allowable
Reference declaring it permissible under certain conditions
Reference declaring it not permissible.
There are ways to study medicine and practice as a physician that conform the commandments according to to Islamic rulings. So, you need not abstain from this career for religious reasons.