Translation:You should go see a doctor.
While this translation works in real life, if you want to be really specific then the 去 (go) should be added as well.
No, "看病" is a verb meaning either "to see a patient" or "to see a doctor"
应该 means should which makes this sentence more of a suggestion.
I feel confused why ".. go see..." can be used in English. "..go to see.." is much better.