How does Overzealous Fan's defend trigger ("When I defend, you may kill me to move an attacking unit to its base") work with Not So Fast under the new rules? Does Overzealous Fan survive if my opponent counters the ability?

No, Overzealous Fan does not survive if countered. Under the Unleashed rules update, "kill me" is treated as a cost within instructions. You must kill Overzealous Fan first (paying the cost) to put its defend trigger onto the chain. Your opponent can still play Not So Fast to counter the triggered ability, but Overzealous Fan will already be dead because the cost was paid upfront. The "may" in the text means you have the option not to kill Overzealous Fan at all, in which case the ability never goes on the chain. Previously, Overzealous Fan would survive a Not So Fast counter because the kill happened on resolution, not as a cost.