3 months after a member's expiry date they become "Expired" and their cards deactivate. See Membership Accounting for more info.
Technically this could be 4+ months after the last payment was made since membership is paid ahead of time. For example if I paid for 1 month today, my cards would deactivate in 4 months from today.
No, Spaceport is only notified when a payment has occurred. It's unable to request or refuse payments from members. It's also unable to manage payment subscriptions. See PayPal Integration for more info.