What will happen if I reach free limits in Google App Engine (GAE)? Will GAE force me to pay for this? Will GAE just stop handling my app requests?
They will not make you pay, although its highly encouraged else your APP might go down or show an error/redirect to users until the next quota cycle (or your manual re-up).
When an application consumes all of an allocated resource, the resource becomes unavailable until the quota is replenished. This may mean that your application will not work until the quota is replenished.