score:3
with my first class ticket for the first leg can I still use the first class TSA lines at airport A or do I have to use the regular lines
Depends on the specific airport: some don't even have a premium security line. It also depends on if you have OneWorld status and if it's domestic or international First (which is a HUGE difference). Ask the agent during check -in to make it happen. TSA PreCheck or CLEAR are better options (if you have access).
Also, can I check my bag in at the first class bag check at airport A even if I'm not flying first until I get to airport B?
Yes. Depending on the airport, there can be multiple premium tiers. International First is always the highest tier. Domestic First without status is often the 2nd highest tier. The signage will tell you.