1. If it is for a shower arm it DOES need support, but you cannot just run it from a water pipe, you need a shower control valve.
2. It is a trap, and that is a water line, so if you are moving the trap you have to do it correctly, not just however it can be done.
3. Unless that is already a drain line to a trap, you may NOT be able to cut into it for your bathtub. And if it is to a trap, then the flow is FROM the trap to the drain system.
These three questions imply that you did not learn enough from that one shower installation to be the "plumber" to do this one. ESPECIALLY since we do not even know if that one was done correctly.