Note that the route is slightly different in Game B, But each trick still can be performed as long as you use the PRG0 version of the game. If you don't use PRG0 version of the game then the right facing jump at the end will not work.

This video explains the spring jump and right facing backwards jump in stage 2 and also shows off the route for stage.

Personally when I jump at the start I let go of A and then start holding it again when I'm about to touch the spring.

For the right facing backwards jump you don't need to do a backwards jump at all if you get to the ending extremely fast.

What you can do is just get underneath the chain at the end that's to the right of the platform, turn right, and then perform a normal jump by pressing A.

But you have to make sure that you're far enough to the left of the chain, yet right enough of Mario (The person pulling the switch sending the birds at you.) so that you actually grab the chain and also don't die by getting touched by a bird before you grab it.