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Yes, a Jennings J22 will fire without a magazine inserted.
These retail for about $85 new.
That's a task best left to a gunsmith.
frame
Best thing is to have a gunsmith show you.
Recommend you ask a gunsmith for help
The thing that keeps the slide on a j22 frame that won't stay on is the slide catch. It is designed so that the slide mechanism can be locked in place without risk of it moving.
Best to ask a gunsmith for help if you don't have a manual
.22 LR rim fire
They were about $60 new, and now about $100 in good condition at a gun show.
You will have to experiment with as many different brands as you can afford to find out which it likes.
$50 to $75. These were very crude, low cost firearms of no particularly high quality.