Not sure if this is what you have or if it might help? You should be able to expand these pictures to get a better look.

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plumb, is your extractor 6 o'clock or 7?

For the rifle’s half cock issue, sounds like the trigger spring might be too weak, or the sear notches are worn or bubba filed to much off, there is a resent post about this. 

For the loose lever the link pins can be worn out, the link itself can be worn with oblong holes for the pins, and the pivot pins for both the breech block and lever can be worn out. Does your rifle have the pivot pins or actual screws for the breech block and lever?
Most model 44’s have a lever bushing that could also be worn or missing causing side to side play. Some model 44’s do have a spring and plunger to hold up the lever, most do not.

The above top two illustrations are for a 44 with a 7 o’clock extractor which is basically the same as most 44’s.  The bottom illustrations are for the 44 1/2 so disregard them.

For the action looseness, I would start with removing the breech block and lever assembly and remove the link pins and inspect the link and pins, then inspect the pivot pins or screws for wear, and replace or repair what is needed. Also check to see if the lever bushing is worn or missing, it’s on the left side of the lever where the pivot pin goes thru. That should tighten up the action.

CPA probably has the pivot pins in stock, but for the link you’ll have to find a good used one, or drill it for oversized pins,and the link pins will have to be made from drill rod or other modern steel.