I have a creature with regeneration. My opponent responds to me activating regeneration with the Turn side of Turn // Burn.
Can I then activate regeneration again to make it a 0/1 weird with regeneration?
Yes, you can activate an activated ability as often as you can pay its activation cost.
But you don't need to. The ability will still resolve even if the creature no longer has the ability when it resolves.
For example, say the creature in question is a Dutiful Thrull.
You could activate Dutiful Thrull's ability a second time between steps 3 and 4, which will give it a second shield for later on in the turn.