MacBook – Does constant charging harm the Macbook Pro

As suggested, SMC reset should be your first port of call.

If that doesn't change anything, try removing the battery and see if the system recognises the change. (Status should change to "no battery"/battery icon with an x). If it changes, that pretty much means that the battery is toast.

I'm guessing that your battery has failed as this is the most likely cause of these symptoms. Cycle counts aren't a sure fire way of determining battery health, they can just die.

If your battery is less than 12 months old, you are entitled to a replacement from Apple.

Deeper discharge/charge cycles shorten an Li-Ion battery's lifespan faster that shallow ones. (e.g. One 100% cycle is more wearing that four 25% cycles, etc.) Therefore, keeping an iPad or iPhone on a charger or always topped off between short uses will tend to lengthen the battery's total lifespan by reducing the number of deeper cycles per annum, all other thing being equal (such as total time of use).

"Calibrating" has almost nothing to do with the battery lifespan. It's mostly needed to help the battery gauge icon behave more accurately. Once or twice a year is probably good enough for that purpose.