Your translation is right, only it seems that county-level hospital is more preferable than county-leven city hospital. I don't know what the system of hospital classification in your country is like (or even, if there's such a systen at all). In mainland China, hospitals are estimated and rated, and then each granted a grade: A1, A2, A3; B1, B2 and so on. This kind of estimation and classification is based on a hospital's medical functions, facilities and technologies, and thus scientific. But there're another even more commonly used ranking 'system' : township-level hospital, county-level hospital, province-level hospital etc. This system is antiscientic and unlawful, but there are tons and tons of such weird phenomena in china: the basis of all the oddity comes from an underlying notion deep in lots of Chinese people's mind: everything can be estimated and rated according to bureaucratic hierarchy. It'll take a long long time to change this backward situation.

县and县级市 is similar, but 县级市 is more urbanized. As you say,县 is like county,but 县级市 is like county-level city. 县医院 is in the county and the 县级市医院 is in the county-level city.