Reddit mentions of The Trinity: An Introduction to Catholic Doctrine on the Triune God (Thomistic Ressourcement)

In addition to the above, if you're looking for something that's maybe more easily digestible (Augustine, while a masterful writer, can be difficult if you're not used to reading ancient texts), you might try The One Thing is Three by Fr. Michael Gaitley or The Trinity by Fr. Gilles Emery.

Gaitley is the more accessible of the two, as his work is largely reflective and filled to the brim with anecdotes, while Emery presents a much more theological approach to the topic.

EDIT: Also, the full text of Augustine's De Trinitate is available on New Advent, if you're not able to purchase the book.

In addition to what has already been said you may like the following book on the Trinity in response to point #1 and just generally:

https://www.amazon.com/Trinity-Introduction-Catholic-Thomistic-Ressourcement/dp/0813218640

It's really good, written by a Dominican so you know it will be clear and thorough, very academic but approachable.

A lot of thoughts here, and please, any specialists feel free to correct me.

Emptied himself.

• We shouldn't understand this as Christ emptying himself of his divinity. His divine nature doesn't increase or decrease. The emptying here should be understood as to the visible manifestation of his majesty (coming as a lowly babe, born in a manger, in time), and to the extent of his obedience and humility for the sake of our salvation (e.g., choosing the way of the Cross).
  
• Christ cannot be separated from the Trinity. The divine essence is singular and does not admit of division in itself. The three divine persons of the Trinity cannot be divided from this singular divine essence of God. Meaning that Christ, in his visible mission (more on this below), is not separated from the Father or the Spirit.
  
  Visible Missions
  
  
• Trinitarian theology (especially from St. Thomas Aquinas) has a principle that the internal, eternal, immanent processions of the Trinity (the Son proceeds from the Father, the Spirit proceeding from the Father and the Son) are known to us principally by their visible missions that were/are manifested and revealed to us in time.
  
• The Incarnation reveals the nature of the Son to be begotten by the Father, technically his procession is one of generation (as distinct from the specific procession of the Spirit). The visible mission of the Son is all that happens from the Annunciation to the Ascension.
  
• Pentecost reveals the nature of the Spirit to be sent by the Father and the Son. It is related to and occurs after the Ascension (the Son sends the Spirit, with the Father). The visible mission of the Spirit was made manifest to us at Pentecost, and still continues to this day through the Church, the Holy Spirit is her very soul.
  
  In a round-about way to answer your question: where one person of the Trinity acts, all act / are involved, because the persons are not divided in essence. So, the Holy Spirit is present in the visible mission of the Son, while he was at work on this earth (Annunciation to Ascension).
  
  I'd really suggest The Trinity by Gilles Emery. It's a short book, and relatively accessible.