Let me begin by listing all four verses you quote with my very (overly?) literal translation:
- Col 1:3, Εὐχαριστοῦμεν τῷ Θεῷ Πατρὶ τοῦ Κυρίου ἡμῶν Ἰησοῦ Χριστοῦ πάντοτε περὶ ὑμῶν προσευχόμενοι, = We give thanks to the God, Father of our Lord Jesus Christ, always praying for you,
- Col 2:2, … εἰς ἐπίγνωσιν τοῦ μυστηρίου τοῦ θεοῦ, Χριστοῦ, = in a knowledge of the mystery of God, [namely] of Christ,
- Col 3:17, … πάντα ἐν ὀνόματι κυρίου Ἰησοῦ, εὐχαριστοῦντες τῷ θεῷ πατρὶ δι' αὐτοῦ. = all in the name of Lord Jesus, giving thanks to God Father through Him.
- Gal 1:4, κατὰ τὸ θέλημα τοῦ θεοῦ καὶ πατρὸς ἡμῶν, = according to the will of God and Father of us.
There are several things about these we should note. First, none are particularly challenging either theologically or linguistically. Second, "Father" does not ever have an article (= "the"). This is mostly the case throughout the NT.
The theological truths taught in these verses are:
- God the Father has this title because He is called the father of Jesus Christ
- Jesus is the mystery of God
- We should give thanks to God through Jesus
- God's salvation is according to the will of God our Father
- God the Father is also called our Father
Not one of these is controversial. Most of the differences occur because the English translation adds "the" and "and" that often do not exist in the original Greek, but are added to smooth out the English.