To avoid dry firing damage, I'd like to put a spent case in the chamber before pulling the trigger. However, if the carbine uses a controlled feed, this may eventually damage the extractor.

It's a good practice to NOT dry fire any rimfire firearm. There are some exceptions to the rule, but just to play it safe, don't habitually dry fire rimfires.

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SIC SEMPER TYRANNIS

The 15-22 has a pivoting extractor that can snap over the rim of a chambered round with no damage. The controlled versus push feed really doesn't factor into this design because it doesn't have a bolt like a Mauser 98.

However, there is a much easier way to do it. Put the orange safety flag that came in your rifle in the chamber. Dry fire with the flag in place. The hammer will strike the bottom underside of the bolt and never touch the firing pin.
Problem solved. Dry fire to your heart's content with no damage.

The factory ships every 15-22 with a dry fire device (the orange chamber flag) but most people don't realize you can use it as that.

The broken extractor problem is basically only center-fire semi-automatics. But no rim-fire should ever be dry-fired because it can damage the chamber/breech face. And the firing pin can break because it isn't cushioned by the cartridge case. JRD is right about using the chamber flag.

Mine never came with a chamber flag!

Now I'm puzzled; the chamber flag will hold the bolt out of battery; doesn't the disconnector keep the hammer from falling under that circumstance.
Need to go look at mine.
Moon