I just bought a model 22F in 20ga and 222. When delivered I opened the action and a little part fell out. Part of the extractor I suspect. The action doesn't appear to cock when opened. I don't know if this is normal. To fire, the barrel selector is slid into position and the rear trigger is pulled to cock the action. Pulling the front trigger fires the barrel. To fire the other barrel, move the selector to that barrel, cock with the rear trigger, and fire with the front trigger. The front trigger release is delicate like a set trigger, so I'm wondering if the gun should be cocking when the action is opened. I'm guessing to a degree, because I don't know how it is supposed to work.

The fore end is removed by taking out an attachment screw holding it in place. The barrels don't tip off like a shotgun. The hinge pin must be removed for that.

Any help would be appreciated.

I presume you have got some more info on the operation of the 22F model. You are correct they require manual cocking with the rear trigger, they do not cock on opening.
The single extractor extracts both the shotshell and the metallic cartridge and retracts into a recess as the gun is closed so should be easy enough to see if the little part that fell out of the gun when you opened it is indeed a part of the extractor broken off. Because it is not an ejector subject to shock of an activating spring or hammer, welding the broken part back on to the simple extractor should be successful.