(Not a Teacher)
2) Obviously, if you are referring to a real patient, you will use the gender appropriate pronoun. However, if you are speaking generally or hypothetically, you can use whichever pronoun you desire, though 'he' is the most traditional pronoun to use in this instance.
"If a patient displays these symptoms, immediately bring his temperature down with a saline IV and prescribe the following antibiotic regimen."
To me, it's a little awkward to use 'they' as a gender neutral alternative when the subject is singular, but I've seen it used enough times to conclude that it's acceptable to most native speakers:
"If a patient wants ice, they can use the machine in the breakroom."
3) Once upon a time, English had noun declensions and gender distinctions. However, over the course of history it lost most of these features. Few words still retain a masculine and feminine form. I believe most words that have this feature were imported from other languages and carried their according gender distinctions with them. For instance, the endings -ess and -ette came to English with French loanwords and were then applied to other English words to denote gender.