When I insert the film, it won't catch, therefore I keep winding until the roll is done (also, the shutter release button won't depress). What does this mean? Is there a way to fix it?

This button you speak of is an alternate way of depressing the shutter it is also a way in which you can use a shutter release cord to take pictures. the button itself is not meant to come out of the camera. Once you load your film or 2 empty film spools (to test to see if the camera is working properly) into the camera and wind the film crank then you will be able to depress that button and fire the shutter.

http://www.butkus.org/chinon/mamiya/mamiya_c330f_prof/mamiya_c330f_prof.htm

From the manual:
"If the error display is not canceled when the shutter release button is pressed again, insert the Dark slide, half-press the shutter release button while pressing the roll film holder's manual film advance button to wind up the film, then contact your nearest Mamiya agent or service center."

If you need a manual, you can download one from
http://butkus.org/chinon/mamiya/mamiya_645af/mamiya_645af.htm

You might want to check and make sure the battery is still good. If you purchased the camera used and didn't change the battery then this might be the case.

But in the mean time this is what you need to do:
1st-make sure that the shutter speed dial is set to any shutter speed other than the red circle w/red ring. If your camera is set to that it will drain your battery when not in use since it is designed to be an automatic setting. I set mine to 1/125th to do this push in the circular dial on the side and at the same time turn the knob.

2nd- locate the 2 levers on the opposite side of the shutter speed dial. Once you found them then position the back lever all the way forward towards the setting for "MULTI" and position the lever towards the front straight up.

3rd- then wind the film crank till it won't wind anymore and depress the shutter, if it is wound and the shutter release is still stuck then skip to the fourth step, if something happened and you heard a click then still proceed to the next step because no the shutter release will be temporarily stuck.

4th-Once both of these levers are in this position there should be a red button just behind the shutter release button on the top right side of the camera.... drum roll please...."Push the red button" what you should hear is a rather loud click, this click is the mirror releasing allowing you to: 1 wind the film crank and take another picture and 2 this also now allows you to see through the lens again (assuming that you have a lens on the camera).

In Conclusion: without a battery that is fully functional in the camera this is how the camera will operate once you get a battery for your camera then reposition the levers on the side of the camera opposite of what I suggested (this will be the normal operating mode for the camera with a battery) then when you wind the film crank and depress the shutter everything will function normally.

Please follow these instructions: 1. Open the camera back 2. Pull the film leader across the red film index mark (red line) 3. Check to ensure film is properly positioned with no slack, then close camera back 4. Fully depress shutter release button to advance to frame 1. Check the rewind lever if it turns counter clockwise when you depress the shutter release button.
Keep on trying this procedure until you get it right. Most of the time, the error is in not pulling the film leader enough to reach the red film index mark.
If you're still having a problem with this, the next best option is to have the camera checked by a competent camera technician.

I guess it is actually an AE1 and not an EOS, as EOS is a digital camera! You will have to scrap the film in the camera by opening the back and removing the film. Take out the film can, then pull the film off the take up spool press the windback button on the bottom of the camera. This should get it working again. Try a few shutter releases and wind ons before putting a new film in. If you have important images on the film you don't wan to lose just do the above in complete darkness, and wind the loose film back into the can using the knob at the bottom.

After taking the first shot, hold down the Multi-Exposure button while cocking the film-advance lever. This cocks the shutter without winding the film. You can continue to do this for as many exposures as you want. Just don't hold down the M-E button while cocking after the last exposure.

The Multi-Exposure button is the button on the front of the camera just below the shutter release button.

Check the release button lock tab, it may be engaged, holding the release button in

nadyne,
manual rewind is next to lcd panel right side. hold the button down for 3sec. the re-wind symbol will be displayed on the lcd panel.while holding down the re-wind button depress the shutter button. film will re-wind.

I am having a similar problem with my eos 100QD.
shutter opens, then stays open until I depress the shutter switch again.
Sometimes about half my photos are lost on a roll, but the other photos are fine (as would be expected from how the film is wound). This problem looks like it's caused by the shutter mechanism failing, a common problem with older camera and one that is often more expensive to repair than it is to replace the body.

there is a flexible ribbon cable carrying signals from camera body to the lense assembly. Due to regular use it gets worn out, thus it gets broken and is not able to perform the task for which the camera has given the signal. in your camera also the same flex cable is broken. Get it changed at some good repairer