How did God become God?
God had no origin. If God became God, he must have been something other than God before that. God does not change (Malachi 3:6). God was always been and is now and will always be (Rev 1:8).
God has always been God. When time came into being and all things became, God already was, and had already been. Nothing exists that was not made by the Father through the Son (John 1:3).
As a logical argument, when we say "nothing exists that was not made by God," this includes all of the cosmos and time itself. Hence, the words "become God" are not meaningful. Nothing can change except for created things, because change (ie from something other than God into God) requires time to have a "before the change" state and an "after the change" state.
Has God always existed?
Yes. God has always been the same and he exists now, so he has always existed. From the answer to the first question, if you mean "was there any time before creation when God did not exist," the correct response is, "there was no time before creation. There was only God."
Does God have a God?
The Son loves the Father and obeys the Father's commands and abides in the Father, and the Father loves the Son and abides in the Son (John 14-15).
The Father refers to the Son as "God" (Hebrews 1:8).
God exalts himself above all worshipped things (Thess 2:5).
God sets himself above all things that are worshipped. He is supreme to all things that are worshipped, and so he does not worship anything created. All things that exist were created by him and through him and to him (Rom 11:36). Then the only worthy recipient of God's worship (or our worship) is God himself. We see this as the Son worships the Father (John 17) and the Father worships the Son (Hebrews 1). Worship of anything other than God, or referring to any being as God other than God is idolatry, which is a particularly grievous sin.