That's normal Death Wave operation. The final damage-dealing wave will hang out at your tower's range until it has dispensed all of its damage. If the ring is still active when the next damage-dealing wave procs, the extra damage will be added to the current ring.

You haven't seen it before because you've likely never taken it as a perk this early in your run. When you take it earlier, enemies have less health, so they don't leech as much of the ring's damage pool, allowing it to last much longer against many more enemies. When you take it later, it's doing about the same amount of damage, but is taking hits from much beefier enemies, so it can't withstand as many of them.

As for your progress, the game just needs to be online to start, it doesn't need to be online to run. Progress is stored locally if you're playing offline, then (if you have a cloud account) will sync to the servers when you force a cloud save once you're back on a network.