Yes. In order to register your vehicle in New York State, you need to have New York State issued automobile liability insurance coverage.
for how long should the liability insurance coverage remain in effect?
As long as the registration of your vehicle remains valid.
Does this apply for motorcycles too?
There are some exceptions for motorcycles.
Regarding the name on my vehicle registration application and the one on the Insurance Identification Card, do they need to be the same?
yes, the names must match in both documents.
Can I submit copies or faxes of my Insurance identification card to the DMV?