Puritan Board Doctor
(1) What precisely do we mean when we say the Abrahamic Covenant, etc., are administrations of the covenant of grace? What precisely does it mean to administer an agreement between two parties to other people?
We mean by administration (or dispenation) a period of time during which the covenant of grace was administered in a certain way. E.g. from the time of Moses to the time of Christ certain accoutrements were added, which you may read about in Exodus - Deuteronomy. From the time of Christ until today we have a simpler administration which you may read about in the New Testament.
(2) What precisely do we mean when we say "Abrahamic Covenant," "Davidic Covenant," and "Christ's Covenant"?
You may read about God's various appearances to Abraham in Genesis, including the giving of the covenant sign and seal of circumcision in Genesis 17. God covenantally promised Abraham land, a people, the Messiah and blessing. The Abrahamic Covenant was administered more simply from the time of Abraham to Moses, then in a more complex way from Moses to Christ (because of the childhood state of the church-nation, and in order to give plenty of types for our own teaching). Now the Abrahamic Covenant is administered in the simple way of the New Covenant.
The New Covenant isn't called Christ's covenant, because Christ was behind all the Divine Covenants.
You can read about the Davidic Covenant in II Samuel 7 and in the Psalms and Prophets. In the Davidic Covenant it is promised that the Messiah would come from the line of David, that He would be King over His people Israel - including Gentile believers, that He would be King over the whole Earth, and that his reign would take the form of a Kingdom.
(3) Related to 2, if all the other covenants are administrations of the covenant of grace, wouldn't they also be "Christ's Covenant?"
Christ was behind all the revelations of the Covenant of Grace, from the Protoevangelium to the New Covenant, because Christ the Son of God and the Word of God (John 1:1).
Even although certain periods of administration are over e.g. the administration of the Mosaic (or Old) Covenant phase of the Abrahamic Covenant, we still - in the New Covenant phase of the Abrahamic Covenant - learn from what the Scriptures say about the Old Covenant, because all of Scripture points to Christ and all is for our teaching.
All scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness: That the man of God may be perfect, throughly furnished unto all good works. (II Tim 3:16-17)