Yes, LED strip lights will carry on working after they've been cut as long as you cut along the designated lines. Cutting elsewhere on the LED strip will cause that circuit, and potentially the whole strip, to cease working.
You can, but we wouldn't recommend it. By cutting an LED strip light anywhere other than the designated cutting point, you run the risk of damaging the components on the strip itself, as well as the circuit board. This may ultimately result in the strip light not working.