Does my film camera actually work if the screen is at 0 but there’s a shutter?
You don't mention which make and model of film camera you have, which is a much-needed piece of information. Film cameras, especially those made in the late 1990s all try to tell the user that 1) the film was not loaded properly, or 2) there's no film in the camera. They do this by not advancing the frame number unless the film has been properly loaded into the camera. Your camera either has no film in it, or the film was not loaded properly, that's why the counter is staying at zero. Do yourself a huge favor and download a copy of the owners manual.
You didn't secure the film properly on the take up spindle.
Did you put a film cartridge into the camera? If so I would say 0 means you did not do this properly and the film is not winding on. Normally film cameras added up to 24 or 36 depending on film roll size. They not do do a count down because you could use either a 24 frame or a 36 frame in the same camera and it would have no way of knowing what cartridge was being used - so they usually did a count up of munger of pics taken.
My 1982 Minolta X700 35mm SLR will happily let you "take pictures" all day long, even if there is NO film in the camera! LOL. Don't ask me how I found that out the hard way! As was said, you only saying "film camera" really tells us nothing. I agree that the number one thing you need to do is download a copy of the owners / users manual for the camera. They can be found online for practically any old camera ever produced. One other thing, it might be a very good idea for you to "sacrifice" a roll of film purely for practice purposes. Practice with loading and unloading the film. If the camera allows it, you can keep the back open and observe how the film actually travels through the camera during normal use. Once you have a good "feel" for the camera and film loading, use, and unloading procedures, then you can start in earnest with actually using it.
It is possible the counter is broken. But really hard to say. Do not open the film as you could cause exposure damage to the film. If you have a dark room facility to test, you can take out the film in a proper dark room and examine and test. Otherwise you would potentially ruin the 3 pictures you took if it is working. Or you can ruin the film in generally from the light exposure. Otherwise as stated, did you load the film properly? Do you know how? What brand/model camera is it? Search the internet to see how you load it if you don't know.
Does it actually have any film?
It seems as if you have not loaded the film correctly, thus it is not moving through the camera when you wind on after every shot. Indeed the shutter clicks every time but since the film has not moved you will not get the pictures.
What camera? Were you the one who loaded film in it? Did you load it properly and advance the film counter to 1 before starting taking pictures? Bear in mind that the counter automatically resets to 0 upon opening the back door. Three shots in may be just right for some cameras to advance to the first frame. It would really help a lot if we knew what camera you are talking about.
The shutter opens and closes mechanically. You will hear the shutter no matter if it has film or not. The shutter opens, the light hits the film and you have photograph. But you will get the 'click' anyway
That is part of the "film" experience, you'll find out when you get it developed. And you'll load the film more carefully next time. I used to keep a roll of spent film to check the exposure counter when I had a question. Thank you for reminding me how nice digital photography has become.
Some cameras (like my 1985 Nikon N2000) have on the back cover a small window and turning indicator that shows the film moving when it advances. But the film counter, if mechanical and working properly, should advance after zero is reached. But in most 35 mm film cameras, after you close the back on the film canister and film tail reaching as far as the take-up reel, you advance 3 frames with lens cap on. Then that wasted light struck film is out of the way. The film counter should be at zero and now you take the first actual picture that should be good. You might get 20 or 24 or 36 frames (maybe one extra sometimes) before the film is at an end and you rewind.
See if you can find a manual for the camera or a close model to it online. It'll tell you everything you need to know about that camera's operation.
The film is supposed to advance the number by pulling on the sprockets. You probably didn't load the film correctly. Find a youtube vid on how to do it.