If the cemetery is private, the owners can tell you to leave. If the cemetery is public there should be no problems.
You can, but I am not sure if they will give it to you or not...it just depends on their policy. Some smaller cemeteries love the idea of this, larger corporate owned ones might not.
Yes, you can ask the cemetery office for this information. If you know someone buried in a location without a stone, you can also create a paper marker and photograph it as well. Here are some examples: