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Nice toy, Scott - it looks like the action parts are worn, most likely the action pins, which can either be replaced (try Wisner's Obsolete Parts:
http://www.wisnersinc.com/model/stevens ... 1894-1915/ ) or simply rotated to present a new bearing surface.
The lever is not supposed to drop free until the shooter pulls down on the action lever.
The lever is held up by the action linkage pushing the breechblock up and forward into the rear face of the barrel (see below, about barrel fit).
After the breechblock makes contact the lever should lift another 1/4" to1/8" and snap up against the bottom of the frame firmly as the linkage cams over beyond center.
The reason the hammer on your gun affects the action latching is that there is a tang on the action link between the lever and the breechblock that pushes the hammer back out of the "fired" position onto the "safe" position when you start to drop the action lever.
It sort of works in reverse in that the friction of the hammer against the link helps keep the action in the locked position.
When you pull back the hammer, you remove this friction and the worn linkage won't keep the action closed.
Many of these older guns have enough wear that the lever isn't firmly "spring-loaded" against the frame, but will stay in contact on their own.
Yours sounds like it has more wear.
Also check the takedown screw to make sure that it is pulling the barrel back tight against the receiver. The screw has a tapered end that fits a taper in the bottom of the barrel extension.
About the rear sight - since that rear sight doesn't match up with any Stevens rear barrel sights I know of (or that I can find in Stevens catalogs from 1898-1930), I would suggest taking the entire sight to a good machinist or real gunsmith, so they can make a replacement adjustment screw/knob.
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