Bogbabe, I don't want to dominate this thread with war stories but to answer your questions;
Keep in mind that usually the person has not been heard from for days. For example someone may be a no call, no show at work for a couple days in a row and they don't answer the phone and someone checked on them and couldn't find them or maybe an adult child calls their parents every day to check on them and get no response for a couple days. This narrows the chances down that something is up. Or we get called by an alarm service.
Yes, I bang on the door, ring the bell, use my personal cell phone to call if I have the number. I listen for the phone in the house ringing to make sure I am calling the right number. I try to look in all the windows for signs of life. If there is a pet in the house I try to see if there is food and water in the bowl indicating that it was recently fed. I look for newpapers and mail that have been there for days. Then, if there is no way I can confirm or deny the safety of the person I will break in, again, doing as little damage as possible. Usually if they have a sliding glass door or a regular window I can get in without doing any damage at all. I also carry a set of lockpicks and if they don't have real good locks I can open the door with no damage. You can be sure that throughout the whole time, I am yelling, FIRE DEPARTMENT !!!!! IS ANYONE HERE ?. If I do break in I stand to the side of the door and yell some more or squat on the window ledge before jumping in. I watch for dogs, especially if there are food bowls outside or dog crap around the yard. I pay attention to the size of the crap also. The situation where I kicked in the door was highly unusual. I did this because the person in question was the mother of someone I personally knew. As I tried to get across, we were led to belive this was a dire circumstance etc. Very seldom would I kick a door. Obviously if this was all a mistake I have caused expensive damage to the house and we now have no way to secure the house so we can leave. If we get a call like this to an apartment we get the apartment manager to open the door with a pass key if we feel time is not of the essence. If we can, we call back the person who called us and ask them if they have a key and can come and open the door. Breaking in is a last resort but it does happen, probably every day at least once somewhere in the city. And yes, I have found people seriously ill, injured or dead when I got inside on a number of occasions. If I can see in the windows and nothing seems amiss I leave. If there is no car in the driveway I leave.
Wrong addresses. Mistakes happen. Maybe the dispatcher misunderstood them. Maybe the person in their panic got confused. A common senario would be a call that was dispatched as 500 North Main Street and it was supposed to be 500 South Main Street. Maybe the person calling didn't even know the right location such as someone staying in a hotel but what room ? Maybe someone called about someone in the Holiday Inn room 110 but there are two or three Holiday Inns in town. Another problem is when you have streets with the same name in two different but adjacent towns. Dispatch gets a call on Main St. so they send the Smalltown FD but actually the call was on Main St. in Bigtown. Most 911 systems have a way to cross check the number with a known address, but sometimes people use their neighbors phone or maybe the neighbor called for the other person. Maybe someones kid was making prank phone calls. Maybe someones kid called 911 and hung up. Maybe they called with a cell phone. There are infinite possibilites.
How often does this happen ? Not often, but when you are running a thousand calls a day, mistakes occur.
Don't forget that all this same stuff can occur with a burgler alarm or a fire alarm. Alarms malfunction frequently. Someone will come to check it out if an alarm is recieved. You may have just burned food on the stove and set off the alarm. Or again, you may have hit the alarm panic button by accident. You may have a silent alarm that you tripped when you came home and forgot about it. As double naught spy said (very nice post by the way), the possiblilies are endless and you will have to make a split second decision. One note, if they are wearing uniforms hesitate for at least a few seconds.