I want to get into analog photography and want to check if my dad's old Canon still works; especially if the shutter is still in working order (as that model seems to suffer from sticky shutter blades). Can I fire the shutter without film?
Yes, you can fire the camera without film.
In general, your test with a film rangefinder/SLR would be to open the back, remove the lens, and cycle and test every selectable shutter speed.
You’re not going to be able to confirm if your 1/30 selection is actually 1/40 or 1/20. You may be able to run a timer for longer shutter speeds to test for accuracy...but even if they are accurate, there’s no guarantee that the shorter speeds are too.
Instead, look for proper function (as you noted, no stickiness).
Then, get a roll of film and test the camera. Figure out a perfect exposure of a subject using a meter or another camera and then roll through every shutter speed that you can on that subject. Keep notes of your frame and chosen speed. Then, check the negs.
You should have the same exposure all the way through. Look for anomalies. Pending the type of anomaly, you can figure out the problem.
If you really, really want to set up the camera to run properly, then get it serviced. You’re looking at ~$250 in the States, assuming just a CLA (more of replacement parts are needed).
For the Canonet Specifically, please see https://camerapedia.fandom.com/wiki/Canon_Canonet_QL_17_GIII
The shutter speeds can be manually set and should be for any testing. Do note the battery advice in the article:
It was designed to use the PX625 mercury battery, no longer available in the US. The circuitry in the camera holds up well against slightly higher voltages, so you can use a PX625A alkaline battery as a drop-in replacement, but you will find your exposures tend to be slightly off. Only the autoexposure system requires a battery, however. The mechanical shutter functions perfectly under manual settings with no battery whatsoever.
Yes, you can dry-run the camera. Note that most (rangefinder) cameras, such as the Yashica Electro 35, will fire at a fixed shutter speed without batteries in the camera.
Testing if the shutter fires at all will, however, not tell if you if the shutter speeds are accurate. After all, time wears out equipment, and oftentimes the shutter speeds start to vary from their intended speeds.
For cameras with a manual mode
A way to test this is to dedicate a (partial) roll of film to testing purposes. Set up a scene, in which the light will not change for the following photos you will take. This is of great importance! ...
For cameras without manual mode (such as rangefinders)
This is a little trickier to do, because with such cameras the exposure is measured automatically, and thus the shutter speed is picked automatically too. One could use the light the shutter lets through to determine the speed of the shutter. By making the scene lighter or darker, you can change what shutter speed is being picked.
You will need a piece of equipment to do this. One way is the Phochron XA, which costs $129 as I am writing this. You can also make these yourself as this webpage demonstrates. A lot cheaper, but also requires some technological knowledge I would assume.
Personally I have tried testing the accuracy of shutter speeds using my desktop microphone and a sound editor to try and determine the sound the shutter makes when it opens and the sound it makes when it closes. The 'distance' (as seen in the soundwave) is the shutter speed, but this method is not as precise as using light.