Is my TC hard disk dead?

I still couldn't even complete the new first backup, backup started successfully, but always failed after copied around 4GB files, with the same error message "error occurred while copying files".

Make sure that there are no anti-virus applications on your Mac, or any other utility that claims to "speed up", "optimize", "keep" etc your Mac.

If the hard drive really was "dead", you would not be able to to back up at all.

If you continue to have the same issue, there may be a physical defect on the Time Capsule disk surface.

How can I perform disk error checking on TC?

Unfortunately, this is not really practical unless you are a dedicated do-it-yourself type of person experienced in small electronic repairs.

You would have to open up the TC, pull the hard drive, install it in a SATA to USB caddy or enclosure, and connect it directly to your Mac to run Disk Utility to see if the application might be able to repair the TC disk.

But....If there really is a physical defect on the disk surface....as I suspect, since you have completely erased the data on the drive......this is not repairable, and the hard drive will need to be replaced.

A repair shop might be able to do this for you, but the cost of a new drive and labor would very likely exceed the cost of a new TC.

and even zero out data (which took 5 hours)

The drive systems today use a system called SMART

It keeps pretty accurate track of bad sectors and errors.

If the drive was faulty it would cause a SMART failure and the TC would give you a bad disk error.

The zero out data is a pretty much telling you the drive itself is OK.

A large number of sectors that cannot be written or read.. it would cause a SMART failure.

Every time you turn on the TC it does a check of SMART .. so you can be pretty assured of it working.

Unfortunately your experience is like many others with Sierra or High Sierra to network target.

My recommendation is to download and try out Carbon Copy Cloner. It is free to use for a month.

If CCC can create a working backup to the TC.. that pretty much proves the TC is fine.

Your issues can be TM itself or the computer networking.