electrical conductivity of ionic compound

To explain that ionic compounds cannot conduct electricity, the expected answer should include two points :
1. There are ions.
2. The ions are NOT mobile.
Therefore, a better answer is :
Ionic compounds cannot conduct electricity in solid state because the ions are not mobile.
Strictly speaking, the answer “no mobile ions” cannot confirm that whether there are ions or not, and thus may not be accepted. (CE level usually accepts such answer because the expectation to the students is not too high.)

Similar, a better answer to the second question is :
Ionic compounds can conduct electricity in molten or aqueous state because the ions are mobile.

Additionally, use “delocalized electrons” instead of “mobile electrons”. Note that all electrons are mobile.