It’s all of the above. Sometimes it’s researching in the form of looking around on sites like printables.com or thingiverse to see if someone else has solved the problem already. Scanning is an option too, but it’s tough to get good results with DIY approaches. The barrier to entry is kinda high in terms of money or expertise. When all else fails, there’s CAD. Computer Aided Design. Make it yourself with measurements and some trial and error :)

Kinda depends, if it’s a popular something, there’s usually a model online that someone else has been kind enough to share (generally on Printables and/ or Thingiverse). My most recent experience with that was the shift knob on my mixer cracked and fell off, a quick download, a few grams of filament, 20 minutes, a few persuading taps with a mallet, and everything was good to go.

Beyond that, it’s a bit of personal preference and a bit of you’re trying to do, something like a dial cluster in a car is going to be far more complex that something like a mounting bracket. The stuff I tend to fix/ replace tends to be fairly small, so personally, it’s a matter of sitting down with a pair of calipers to measure the object and replicate it in CAD (Autodesk got me young, so I’m on the Fusion 360 train at the moment). One of the most amazing things about 3d printers is that you can go from design to prototype extremely rapidly, which allows you to iterate the design and make it better each pass. Got a hole doesn’t quite line up, a wall that’s too long, an arm that doesn’t quite reach, etc? tweak it and try again. It’s a little bit of trial and error, but with experience it becomes more of a controlled process as you figure out what works and what doesn’t.

Usually just taking the broken piece, measuring it out and replicating it. A good pair of calipers helps.

Of course if it’s common enough, you can usually just find things already engineered for your needs.