How many gods did Paul believe exists?

An infinite number, one God per universe.  For us in this universe, there is only one God, the Father, according to St. Paul.  Jesus is only one of the Lords, or lesser divine beings.  The Holy Ghost is only a ministering spirit.  Nowhere in the Bible is he called a god.

Paul was both a Jew and a Roman Citizen, so he acknowledged both the Jewish God and a whole bunch or pagan gods as well..

Lowercase "g" gods can be translated as divinities or authorities. The evil divinities are always called "demons", the benevolent ones are usually called "angels". The former act on the behalf of Satan/Great Dragon, the latter usually act on the behalf of God. It is said that there are multitudes of angels in heaven, so it can be assumed there are proportionately as many demons (at the time of the fall, Satan is said to have swayed 1/3rd of the heavenly host to his side).

One.  Paul believed the Triune God, and ONLY the Triune God--Father, Son and Holy Spirit--exists independent of human imagination.  And of course Paul was correct.

"...at that time the Israelites worshipped him [Yahweh] alongside a variety of Canaanite gods and goddesses, including El, Asherah and Baal, but in later centuries El and Yahweh became conflated..." (Wikipedia).

Try this

1 Corinthians 8:5 — There may be so-called gods both in heaven and on earth, and some people actually worship many gods and many lords.

Note the lowercase g in gods and l in lords.

And the Hebrew word for gods in Psa 82:6 means judges in this context.

God stated early on that his faithful Hebrews would judge both the world and the fallen angels.

1 Corinthians 6:3 — Know ye not that we shall judge angels? how much more things that pertain to this life?