The barrel should be tight when done, if it turns past then it needs to be machined to carry on to the next rotation. These things aren't as easy to work on as one might think. There's many small bearing parts that affect the function. Yes, the strain screw should be tight and when adjusting for a lighter trigger, it's a job for someone that's done lots of them. Then you need to use Federal 100 primers. The pin you seek is mild and a standard size. You should be able to find correct parts in Numrich for those.