When a WWE figure bounces another figure off the ropes (or hindering terrain),do they attack again? Otherwise what’s the benefit?

They do if they have this power showing:
SLINGSHOT – (Purple Speed) – KNOCKBACK, but you may choose the direction and can knock back characters that can use Charge or Combat Reflexes. // FREE: Make a close attack targeting an adjacent character that this character knocked back this turn.

So if my speed is purple,I can do a normal close attack,it will knock them back (without rolling doubles?) then when they come back to me,I can do another attack?

BTW kind of stupid to call it “slingshot “ since there is a wrestling move called slingshot,and it’s not that.

To the first part: Yes. the "KNOCKBACK " keyphrase (when it's in all caps like that) is a passive ability that allows the character to knockback any target on any attack (Force Blast also does this, as does Super Strength, but only for close attacks)

You hit them, they knock back into ropes, and after the knock back is finished moving the target, if you're adjacent you can do the FREE: make a close attack. (Also note, they don't actually have to be moved to a different square. You can knock a character into blocking terrain and they stay in the same spot and you get the free attack.)

I'm sure it's called Slingshot because for a non wrestling fan "Slingshot" invokes this type of slinging action. Whereas "irish whip" doesn't really mean anything to me.