Hello,
Welcome to Microsoft Community.
Please allow me to explain to you all the information about the Surface warranty.
Generally speaking, the limited hardware warranty of Surface series devices is one year. If you purchase this device along with an additional extended warranty, the warranty service of your Surface will be extended.
To check the warranty status of the Surface, it is indeed necessary to complete the "Register Device" operation first. So, that's right. It's the "Register Device" button on the dashboard of your personal Microsoft account. Then, enter the corresponding serial number to associate the device with your Microsoft account and simultaneously check the warranty status of the device:
Register your Surface or Xbox - Microsoft Support
How to get service or repair for Surface - Microsoft Support
As for extending the warranty period, Microsoft does have relevant information, but it can only be extended when the warranty information is incorrect. In simple terms, if the extended warranty was not added to the warranty when you purchased the device, you can contact the Surface technical support team and provide them with your purchase vouchers to correct the warranty period of your Surface. For more details, please refer to here:
Update your warranty with your proof-of-purchase - Microsoft Support
So, if you are currently trying to find out if there is a way to extend the warranty period of your Surface, I can only tell you that the prerequisite is that you also purchased the extended warranty service when you bought this Surface.
However, if not, it means that the warranty period for this device has ended. If it is to be repaired by the Surface technical support team, additional payment will indeed be required. To be honest, it is indeed a little expensive:
Get service for your out of warranty or damaged Surface - Microsoft Support
How much does out-of-warranty service cost for your device or accessory? - Microsoft Support****
Now, if your Surface is in a situation where it can start but cannot enter the system, considering that you have already performed methods such as "forced restart", the suggestion I can offer you is to create a Surface dedicated USB recovery drive on another computer and then use it to restore your Surface to factory Settings:
Creating and using a USB recovery drive for Surface - Microsoft Support
As long as your Surface has no hardware failures, this solution is usually effective.
Best Regards,
Lizzie-Z | Microsoft Community Support Specialist