Sounds like you are doing fine. Being smart, too! You are right to not use those pucks. You do NOT want copper in your pool!

I think you are fine to swim. We swam the day our pool was installed.

I would just use bleach for your chlorine. You have already added CYA and chlorine tabs have CYA in them. You don't need more. Stay away from HTH products. They are not good quality and sometimes have a bunch of unwanted junk in them. As Ben says, HTH has gone over to the 'dark side' and you're better off not buying their stuff.

Keep us posted how it is going and let us know if you have any further questions. Enjoy the pool!