This question covers a great deal of ground, so I will only be able to give a short answer to each of the points.
To be self-existent means to be the source of your own existence; to be the source of your own life.
Exodus 3:14 And God said unto Moses, I AM THAT I AM: and he said, Thus shalt thou say unto the children of Israel, I AM hath sent me unto you.
Here, God identifies Himself as the I AM. That title refers to His self-existence. He is of Himself without any other. Here are some other verses:
- Deuteronomy 32:40 For I lift up my hand to heaven, and say, I live for ever.
- John 5:26 For as the Father hath life in himself; so hath he given to the Son to have life in himself;
Notice that the Father has life in Himself. His life does not come from an external source. It comes from within Himself. He is self-existent.
- Psalm 90:2 Before the mountains were brought forth, or ever thou hadst formed the earth and the world, even from everlasting to everlasting, thou art God.
- Isaiah 57:15 For thus saith the high and lofty One that inhabiteth eternity, whose name is Holy; I dwell in the high and holy place, with him also that is of a contrite and humble spirit, to revive the spirit of the humble, and to revive the heart of the contrite ones.
I believe these verses are self-explanatory.
- John 4:24 God is a Spirit: and they that worship him must worship him in spirit and in truth.
- 1 Timothy 1:17 Now unto the King eternal, immortal, invisible, the only wise God, be honour and glory for ever and ever. Amen.
That He is "invisible" shows him to be Spirit.
This one involves much study, so I will only touch on it. Here are some verses:
- Matthew 28:19 Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:
- John 10:30 I and my Father are one.
- 2 Corinthians 13:14 The grace of the Lord Jesus Christ, and the love of God, and the communion of the Holy Ghost, be with you all. Amen.
- 1 John 5:7 For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.
The doctrine of the trinity is built on two facts:
- There is one God.
- The Father, Son, and Holy Ghost are all called God and have all the attributes of God.
Therefore, God is a three-in-one entity: one God in three persons. The Father initiates. The Son executes. The Spirit enables.