Is our civilian police allowed to work together, train, or hold joint exercises with our military? Aren't they supposed to never mix like that? Does our Constitution allow it?
An exception to this would be National Guard units while under the authority of the governor of a state.
The U.S. Constitution allows it. It's only a federal law that prohibits it.
Your question is very difficult to answer, because both the state and federal governments have over-stepped their bounds significantly.
Our Founders very much feared a standing Army. They had just had to pledge their lives, their fortunes, and their sacred honors to over-throw their own tyrannical government, defeating a significant standing army of Redcoats to do it.
Both the federal armed forces AND almost anything that passes for a city police department these days would be viewed as standing armies. To oppose abuses by such, the people were to be protected in their right to keep and bear arms to protect themselves, their families and their communities.
To address the natural concern of the people, there is a law called the "Posse Comitatus Act" which limits the use of the military for non-military purposes. I vaguely recall that some change was made recently having to do with guarding our borders, but I don't have any details.