do left over successes stack out to the next round?

The rulebook is a little unclear on this, but yes, left over successes do carry over.

thorgrim said:

Then what does it mean in the rulebook when it says: "...remove one doom token from the Ancient One's doom track and reset their (the players') cumulative successes to zero "?

It's incredibly badly worded: I think this may be the most common question asked here... It does makes sense if you imagine rolling each combat die individually and evaluating its effect - if any - before rolling the next one.

If you haven't read the AH FAQ yet it answers this question and a lot of other common ones too.

This is specifically clarified in the FAQ in the Dunwich rulebook. Successes do carry over. The alternate rules suggestions (found on this site under Support) have an optional rule that makes them NOT carry over.