Considering Thou shalt have no other gods before me and that the Bible contains descriptions of worship of other deities, combined with the fact that at least the OT God/god seems to be a strictly regional character, are there any references in the Bible that recognizes other gods?
Not that I know of. At my school, we’re taught that God is the ONLY God. While people may consider an object of worship (ex. money, a celebrity, etc.) a “god”, I don’t think there are other mentions of deities or a spiritual/human being that is equal to God. Now, the Trinity is something different because in essence, God the Father, Jesus Christ the Son and the Holy Spirit are all the same God. They are not separate deities unto themselves. There are also angels and demons that are mentioned frequently in the Bible, although, I don’t know if they’re considered deities. Hope that helps. 
Most evidence suggests that the early Jews were polytheistic who went from worshipping many gods, to worshipping their one god who was Better Than the Others, to worshipping the one God who was the Only God. Hence the references in the 10 commandments.
I think other indications of this are in the constant use of ‘us’ and ‘we,’ but I know a lot of Christians who just say that God is talking to Jesus when he uses those terms.
Dueteronomy 4:39 would say “No.”
In traditional Jewish thought, the commandment of not having any other gods has two applications:
(1) One cannot worship another being as a deity.
(2) One cannot worship another deity in conjuction with God (as in a partnership or a member of a pantheon).
Zev Steinhardt
By the time of the prophet Jeremiah (late 7th century, early 6th century B.C.), Judaism considered the pagan idols to be illusions. Jeremiah 10:3-5, 14-15,
It’s pretty clear from the accounts in the Torah that the Jews started out henotheistic, and took YHWH as their own God, He being bigger and badder than anybody else’s god(s) or goddess(es); another term of for this practice is “ethical monotheism”. Note, for instance, Exodus 3:1-15, where YHWH identifies himself not only as “I am who I am” but is also referred to as Elohim/Elohi repeatedly, suggesting a conflation with with El, for instance, and hearkening to a polytheistic bacground. Later, in the Tanakh, you get accounts of showdowns between God’s spokesperson and those of other gods (take the encounters between Elijah and the priests of Baal), which are strongly suggestive of not only Baal’s lack of power, but lack of existence as well. There appears to be an evolving change in the concept of YHWH reflected in the age of the sources.
It depends on the specific book of the bible. If you are considering the overall tone as it exists today (and has existed for 2,500 years, or so), then the message is that there is only a single God and all other gods are either the creations of their worshippers or are otherwise illusory. However, there are certainly passages, particularly in the earlier works, that indicate henotheism rather than monotheism. (Henotheism asserts that our God is the best god, but that there is a chance that your god might exist.) When looking at the passages in which the Hebrew God acted through his prophets to overwhelm the prophets of Egyptian gods or the Canaanite gods, the stories pretty clearly indicate that there was some sort of entity that the other prophets served. However, by the time of the return from Babylon (or possibly before then, based on passages in First Isaiah and his contemporaries), the notion that only the Hebrew God was real was firmly established.