Who do Trinitarians believe is Paul's God?

Constable's commentary in the NET Bible sheds light on the first verses you quote from 1 Corinthians 8:

For instructed Christians there is only one God and one Lord. Paul did not mean that there are two separate beings, God and Lord. These are two names for the one true God who exists as Father and Son. The Scriptures establish the deity of Jesus Christ elsewhere (e.g., John 1:1, 14; 10:30; Col. 1:15-19; et al.). Paul did not argue that point here but simply stated the Son’s equality with the Father within the Godhead.

The point of difference is this. The Father is the source and goal of all things whereas the Son is the agent though whom all things have come from God and will return to God. Since Paul’s point was the unity of the Godhead, there was no need to complicate matters by referring to the Holy Spirit here.]1

As for the verses in Romans 13 and Ephesians 4, Constable's comments shed light on your question.

The Lord Jesus Christ is in no way less than God the Father. The same can be said of the Holy Spirit. The Spirit, too, is fully God (see Acts 5:1-4). (I'd like to think that Paul would concur with the apostle Peter that Ananais and Sapphira lied to God, the Holy Spirit, and not to a man.)

God's person-hood, as it has been revealed in holy Scripture, can be likened--in an always less than perfect analogy--to the three branches of government in the American system: Executive, Legislative, and Judicial. All three branches are in agreement regarding constitutionally derived concepts. Each of the three, however, has different functions.

Unlike the American system of government, there does not exist a system of "checks and balances" within the Godhead, since each person of the Trinity is and functions according to His role, in keeping with the Counsels of God in eternity past.

I will not at this point attempt to link the Father, Son, and Holy Spirit with a corresponding branch of government. I will, however, suggest that there is an agreement, a unity, and a coherence in the essential deity of all three persons, the differences in their roles in the outworking of God's plan for the ages notwithstanding.

Perhaps you should meditate on several passages in this regard (in their respective contexts, of course):

1. Ephesians 1:1-11

2. Revelation 13:8

3. Ephesians 3:11

4. Acts 2:22-23

Paul's God is the God of the OT

Trinitarians of all stripes (Catholics, Protestants, Eastern Orthodox) will interpret the "God" mentioned in all the books of the New Testament to refer to the one God described in the Old Testament. This means when the authors of those books (Paul, Peter, Matthew, John, Luke, etc.) wrote "God" they have in their mind the single most supreme being unequaled in existence who created all that exists,

• the being who made covenants with Noah, Abraham, Israel, and David
• the being worshipped by Job
• the being who sent all the OT prophets
• the being who engineered the 2 exiles and arranged for Cyrus to bring them back,
• etc.

By the time the New Testament books were written, all schools of early Judaism were already unanimous in their beliefs of monotheism and that their God is above all other gods, angels, powers and principalities.

Thus, this is the same answer to your other question since Paul's God is the same being as Peter's God.

Paul's Jesus shows development toward identifying the human Jesus with God the Son

Please don't confuse the slightly different Christologies of NT authors as though they are contradictory. From Trinitarian perspective, it is best to interpret each book's Christology as putting a complementary spotlight on who Jesus is. Thus each Christology is a step toward later, fuller, and authoritative Trinitarian Christology that was agreed on at Chalcedon in AD 451.

But even in preliminary stage, we can already discern agreement among the apostles & authors of NT that Jesus is the being identified in the OT as:

• the Son of Man in Daniel,
• the stump of Jesse,
• the lady Wisdom in the Proverbs,
• the Son of God conceived miraculously within the virgin womb of Mary,
• the suffering servant in Isa 53,
• etc.

which taken together would be critical in constructing a Biblical full-blown Trinitarian Christology couched in philosophical terms ("being", "person", "hypostasis", etc.) a few hundred years later.

Paul's understanding of God as the Father of our Lord Jesus Christ

In Paul's letter to the Romans, we see very clearly how Jesus Christ has a relationship to His Father who raised him from the dead (Rom 6:4, Rom 6:9) with the practical exhortation to us that we should join Jesus in his death so that 1) we live a new life that says "no" to the slavery of sin; and 2) we will also be raised to life as Jesus was. It is very clear that it will also be God the Father who will raise us from the dead, and that we will be declared righteous because we have been united with Jesus.

How do Trinitarians work out the relationship between God the Father and Jesus? I'm going to now interpret the 3 verses you quoted in a Trinitarian way:

1. 1 Cor 8:6: One being (the Triune God) is both our Father and our Lord since the human Jesus Christ is in hypostatic union with God the Son.

   • When we pray to the Father we think of this one being as our creator.
   • When we pray to the Lord Jesus Christ we think of this one being as God's Son who incarnated in our fleshy world to save humanity, who by virtue of Jesus being the second person of the Trinity was also involved in creation.

2. Romans 15:6:

   • Jesus in his human nature calls his God "Father" referring to his conception in Mary's womb.
   • Jesus in his divine nature is the eternal generation of the Father within one being, thus the other dimension of Jesus's intimate relationship with the Father within the Trinity. See GotQuestion article What is the doctrine of eternal generation and is it biblical?

3. Eph 4:4-6: This is a prototype of the later fully-fleshed-out doctrine of the Trinity, because the 3 Persons Paul identified here (One Spirit, One Lord [Jesus], One ... Father of all) was later formalized as the Triune God (one being in 3 persons).