I think the Do = an auxilliary verb and the you is the subject of the sentance. the 'mind will be the main verb.
But then what is the "if I go first part". Is "I" the subject and go- what type of verb is this?
You don't really need help, your analysis is correct.
"if I go first" is a subordinate clause, with "I" as subject and "go" as verb, "first" as adverb."
Try a more complex form with the same meaning:
Would you mind if I were to go first?
Would you mind if I went first?
In these cases, we'd have "go" in the subjunctive. There are only a few traces of the subjunctive in modern English, but it does crop up from time to time.
Can I assume that "if I go first" is an adverbial clause since it answers the question "go when?"
Regards
‘If’ functions principally as a conjunction and as a noun. It’s not an adverb and can’t form an adverbial clause. It’s more a subordinating conjunction that joins a clause (if I go first) to another ( Do you mind) on which it depends for its full meaning. It may further be classified as a conjunction of condition when according to its meaning.
You have a complex sentence here. The independent clause you already understand. The subordinate clause: "if I go first." "I" is the subject, "go" the verb (first person singular of the verb "to go") and "first" is an adverb. That leaves "if," the subordinating conjunction that links one clause to the other.
Need help with analysis 
Linear Mode
