Hi Laura. In my opinion if you are looking a turístic place near Kennedy space center I think best option is Cocoa Beach. There is many options on hotels.
Not real close as it is a huge property surrounded by protected wildlife reserve. However, along Hyway A1A... there are lots.
No
Depending on your definition of 'near'...yes & no. There are multiple hotels in the area, but none within say walking distance. Kennedy is an Air Force Base and a launch site for very large a rockets so the Center itself is quite remote. However within 20 minutes by car/taxi/Uber there are several lodging options at all different price levels.
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