We still say "hoast" to mean "cough" in Scotland!
Why is "she coughs much" wrong?
I don't know, but I believe "She coughs a lot" should be accepted.
In correct English usage, ending the sentence with 'much', 'a lot', and 'a great deal' are all good translations.
she coughs too much wouldn´t be okay too?
Too much = för mycket or alltför mycket