Are the ambulance allowed to enter your home without permission ?

Police would be brought to break into the home most likely.

In situations when people can't get to the door paramedics do sometimes break through the door.

Refusing entry to medical responders will result in immediate arrest for obstruction and disorderly conduct when the police arrive.  It is generally a crime to interfere with emergency services.  Even if "it's a mistake", they have a legal right and obligation to enter and confirm for themselves.

Even with your permission driving an ambulance into someones home is frowned on.

The EMTs probably call for police backup. 

With the limited research I've done on the subject, you can refuse an ambulance ride as long as you're alert and oriented and answer questions appropriately.  If you can't answer questions or if your orientation is questionable, you'll be getting a ride to the hospital.   Be aware if you do refuse to go to the hospital, you'll still be responsible for any bills that are generated. 

The police would come and they would enter.  They have to check out 911 calls.

If they do they will meet a trigger happy Vietnam Vet.

They need to call the police if they believe someone's life is in danger. Say "send police and ambulance" if you have reason to believe there might be a problem getting in.

If they can't get access they will contact the police to break down the door. In this case, the police will most likely already turn up with the ambulance.