What if any, has to do, with His second coming, "more excellent" inheritance? what inheritance?

The inheirtance that Jesus Christ has, is ALL that the Father has. It is to become like God with all His power authority and Joy. It is worlds without end. All the faithfull will receive the same inheritance through Christ and become coinheritors with Him. The purpose of earth life is for us to become like God and receive all that He has.

"He that overcometh shall inherit all things; and I will be his God, and he shall be my son."  Revelation 21:7

"The Spirit itself beareth witness with our spirit, that we are the children of God: And if children, then heirs; heirs of God, and joint-heirs with Christ; if so be that we suffer with him, that we may be also glorified together." Romans 8:16,17

"And for this cause he is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance."  Hebrews 9:15

It's not about his "second coming" at all.

It's about things already passed.

See: the context.

https://www.biblegateway.com/passage/?search=heb+1...

What is better: the "name" (read: "reputation") of Jesus

compared to that of angels.

How: by inheritance. (Jesus is son of God.)

>> What is the common reality of man as inheratance <<

The question is nonsense.

>> Has He done this on earth as it is in heaven? <<

The passage only teaches us that it is so in Heaven, that it is so AFTER "he had made purification for sins".

We can argue that it was immediately after his death

or immediately after the resurrection

or whatever

(the passage does not teach us explicitly).

Simple.  1 Peter 1:3-4.  "A living hope through the resurrection of Jesus Christ from the dead. [4] "to an inheritance incorruptible and undefiled and that does not fade away, reserved in heaven for you."