We divide the patients according to the doctor.
(referring to a method of classification)
We divide the patients among the doctors.
(referring to a method of distribution).
2. b is not English: by-phrases denoting means of transport are invariable. You could, however, say
I went there in a taxi.
3. b is unidiomatic, verging on unacceptable: for nondeictic reference, 'next' normally changes to 'the following':
--> I am going to do it the following week.
(meaning 'the week after the week to which reference has been made').