Does monthly Apple care coverage ever expire? And What is the monthly cost of Apple care coverage for a new IPad

If you're talking about extending the Apple Care Plus protection plan, see this:--> Extend your AppleCare+ coverage - Apple Support

If you meant to subscribe to a new Apple Care Plus Plan, see this:--> https://www.apple.com/support/products/ipad/

Regards

Giulio

AppleCare+ costs an additional $69 (or $3.49 per month for two years) for the 9.7-inch iPad or iPad mini, and $129 (or $5.99 per month for two years) for the iPad Pro and must be purchased within 60 days of purchasing your iPad; most purchasers buy it when first purchasing an iPad.

Also, the coverage can, also, be extended toward the of initial paid coverage.

Extend your AppleCare+ coverage - Apple Support