The OP purchased I fare. @:-)

I purchase I fares ex-TPE several times a year. It is always restricted to same day connections at HKG only. If you want to overnight <24 hours, than you need to buy up to D fare which is a published fare. The only thing OP can argue for is for CX to honor original ticketing error. The overnight was never supposed to be allowed by the I fare rules. It is not a tax or surcharge to overnight, you simply need to buy up to D fare. @:-)

Yes, you can do TPE turn in 60 minutes. Some of the TPE-HKG flights are operated from Japan/Korea. You may want to make sure that you are on the same aircraft for HKG-TPE-HKG. @:-) If not, and your inbound to TPE is late, your return flight may leave without you as the aircraft is different.