Question: Was Jesus God or the Son of God–did God beget him or was he always around–and did he (Jesus) create the world?
All of that is true. Mark 1:1 says Jesus is the Son of God. John 1:1 says Jesus is God. John 1:3 it says nothing was made without Jesus. Colossians 1:15,16 says Jesus is the firstborn of all creation and that in him all things were made. John 1:18 says Jesus is either “the only [begotten] God” or “the only [begotten] Son” depending on what the correct text is. And I put the word “begotten” in brackets because there is a question as to the meaning of the Greek word used here, monogenes.
The word monogenes could mean “only begotten” or it could mean “one and only” or it could mean “unique.” It is used of Jesus in John 3:16. But it is also used of Isaac in Hebrews 11:17. Was Isaac Abraham’s only begotten son? No, he had sired Ishmael earlier. But Isaac was his special son by Sarah, the one to whom he was giving his inheritance. Is Jesus begotten by the Father, or does this term monogenes simply designate him as the unique Son of God as opposed to all created beings who might be designated sons of God?
I lean toward the view that monogenes, when applied to Jesus, means “unique” Son of God. However, Scripture also says in 1 John 5:18 of Jesus that he is “born of God.” There is a doctrine that has developed from this called the eternal generation of the Son, which it says is “an eternal personal act of the Father, wherein, by necessity of nature, not by choice of will, He generates the person (not the essence) of the Son, by communicating to Him the whole indivisible substance of the Godhead, without division, alienation, or change, so that the Son is the express image of His Father’s person, and eternally continues, not from the Father, but in the Father, and the Father in the Son.” (See Theopedia)
What this means is that from all eternity God the Father has been in a relationship with the Son by which He has generated the personality of the Son (and He and the Son have “generated” the personality of the Spirit) so that they share the same essence (deity, divine nature). This makes them entirely equal in every sense of the word so that each is rightly called God, and yet Jesus can also rightly be called the Son of God. This doctrine makes a lot of sense of the data of Scripture concerning Jesus absolute deity (John 1:1) and yet his submission to the Father in all things. This makes it reasonable for him to be the one who takes on human nature (the Father and the Spirit did not do this) and to rule God’s kingdom until it can be handed over to the Father (1 Corinthians 15:24). Jesus is thus “the radiance of God’s glory and the exact representation of his being” (Hebrews 1:3), which seems to speak of some kind of derivation from the Father, and yet at the same time exact equality.
This is a difficult concept to wrap our heads around, but then God is the most amazing and unique being of all, infinite and beyond our ultimate ability to comprehend, yet able to correctly reveal Himself to us in true ways that enable us to know Him.
So the answer to all your questions is, “Yes.”