Yes. It is a US National Monument. National Monuments and National Parks are both managed by Dept of Interior. An America the Beautiful Pass provides annual access to all Dept of Interior public lands - or you can purchase a day pass at Lava Beds NM.
Yes it is a National Monument...
Yes, a national monument
Yes National Monument administered by the National Park System
It does indicate that it is a National Park. I don't know about a national monument.
Thanks! Your answer is awaiting moderation.