Can I check the phone details at the carrier?

If you check with the carrier, you can check the call details when you called.

Can the carrier check the contents of the call itself?

3016 2016-06-04 Gentleman

Call history is possible, but call content is not possible.
Because voice data is encrypted wireless data, current technology does not allow users to view content even if they are hacked.

Gentleman 8 years ago 1

A. 2

The call history is managed by the carrier.
It won't be possible to get to the call.
If you want to save the call, you'll have a lot of data
Now, it's about decrypting encrypted data, but there's a lot of risk of hacking.
The carrier doesn't have to do such a complicated job.