That's only if they don't /cant run the entire DC of battery -whilst the gensets come online this is how telco exchanges / central offices work and important ones may have power supplied via different routes.
Some DCs do, but the need to keep them spinning 24/7/365 is a big maintenance issue, so given the advances in battery tech, I'd assume batteries would supplant flywheels for cost and safety reasons.
For the most part, but every few years the bearings will need replacement. This means shutting the entire thing off and dismantling it for several hours. For redundancy, you therefore need multiple flywheels to cover for the maintenance periods.
Batteries can and indeed do go bad without warning, but current sophisticated UPS monitoring systems are able to detect failures, track battery ages and raise alerts (whether anyone actually acts on those alerts or not is another matter...). I've also found that lead-acid batteries last longest as long as there is some amount of power continuously trickling through them - I have an APC UPS at home with its original batteries from 2010, and they still provide ~1h of runtime at 20% load. And with most DC-grade UPSen, batteries can be replaced in situ without powering the device or any connected equipment off.
As of about 10 years ago our data centers always had 1 out of 4 generators configured with a big M/G flywheel on the output shaft. The flywheel momentum guaranteed generator start, I don't know the latency but it was on the order of a handful of cycles.
Some of them do, but the overhead is ridiculous. I don't have the data in mind, but I think that the NoBreak brand consumes 20kW+ just idling. It's ridiculous
Only some of them have flywheels. The large telco datacenter I visited had huge pallets of batteries to cover the time needed for the diesel generator to start.