Items 1 & 3 may be caused by worn screws and/or worn link. If hammer is going to full cock then the sear or full cock notch is worn, if it's going to half cock then check if the lip is broken. These rifles can be changed to either go full or half cock by reversing the lever/breech block link. Your choice. If you remove the screws to check please be advised that the screw slots on the left side are only for aligning the pins for returning home when replacing in frame, it's the right side screws that allow you to dis-assemble rifle. Many of these guns have been screwed up because the left side will NOT turn. Good luck.

Something to think about and be aware of, some times people put the screws in backwards. Don't apply a lot of force to the screw head, if it don't want to move try the other side. I had one come in last week with the screws in backwards.
If the firing pin is hitting low it could be caused by ware of the pins or link. If the hammer is dropping all the way to the firing pin then the hammer safety notch is most likely broken or the trigger is broken. Also check to see if the barrel has a good fit. Stevens barrels are not interchangeable. If your barrel came off a different rifle it may not line up correct or headspace correct.
Some 44s have a screw in the breach block that will allow it to lock up better, holding the lever up. Some have spring and plunger to help hold the lever up.

MikeT,

What I am going to describe is a rarity, but with all the parts changing that has occurred over the years, it is a possibility. 

If someone installed a replacement trigger with a longer than normal sear tip, the regular lever action motion may not be rocking the hammer back far enough for the trigger sear to drop into the full-cock notch. If this is combined with a hammer that has had a set trigger fly installed, the hammer will go all the way down when closing the action. 

I repeat that this would be rare, but I mention it because I've encountered it.

Regards,
BP