Even if live rock is identified as fully cured, there will be some things that die off in transport to your tank. You'll still need to check for ammonia and nitrites. A lot of times, with fully cured rock, you can skip the cycle altogether, because there isn't enough die off to do anything. But you don't know that for sure unless you are testing for ammonia and nitrites.
If you add the rock to your tank, wait a week, and after a week, everything is still at zero, then you can assume that having fully cured rock skipped the cycle.
You can add a small clean up crew as soon as the tank finishes cycling. You will get several algae outbreaks at this point -- that's totally normal for a new tank. The CUC will help with the algae. Don't add too many at once though. A lot of people add a ton of snails and crabs, only to watch half of them starve to death within a week or two because they've quickly eaten all the food in the tank.