While the devil can lie, He cannot lie to Christ for He knows everything, even the thoughts of the devil.
if you carefully look at what the devil says, the authority "has been delivered". However, if you look at Gen 1, the authoirty over the earth was originally given to man not the devil. In Isa 50:1, God says, through our iniquity, we sell ourselves. Sell what? ourselves. When we sell ourselves, we become slaves to devil and allow his rule. But Jesus came to redeem us from the rule of the devil to restore us the authority over the earth. This will happen when He comes again.
I didn't miss it. You are ignoring it because, even before Christ is given all powers in heaven and earth, God the Father was already having it all the time. Now, Christ seated as the right hand of God, the devil is required to report to Christ who is seated at the throne of God. Christ is waiting for His marriage when He will be one with His children, after which He will put an end to all worldly authority by taking control of it Himself from the devil.