"Researches" is not ok. There is no such word. There are searches, and researchers, but no researches.
A term I often see on billboard ads in Poland meaning also 'medical examination'
1) praca wykonywana przez naukowca, której celem jest uzyskanie nowej wiedzy - work done by scientist which's purpose is to find new knowledge
2)a sprawdzenie stanu zdrowia pacjenta checking patient's state of health
2)b wynik sprawdzenia stanu zdrowia pacjenta result of checking patient's health
Apparently "must" and "should" should be used interchangeably all over the course...
"should" is not accepted in this sentence. Sometimes it really makes sense for "musieć" (hard to discuss without a specific example), but not here.
Depends on the context... but it could be a medical doctor, true. "tests" work.
The 'research' interpretation is probably coming from some scientist.