You asked: "In 1 Corinthians 6:9-11, what does "in the name of the Lord Jesus Christ and by the Spirit of our God." mean?"
Acts 4:12
And there is salvation in no one else, for there is no other name under heaven given among men by which we must be saved.”
Zechariah 13:9
And I will put this third into the fire, and refine them as one refines silver, and test them as gold is tested. They will call upon my name, and I will answer them. I will say, ‘They are my people’; and they will say, ‘The Lord is my God.’”
Philippians 2:10
So that at the name of Jesus every knee should bow, in heaven and on earth and under the earth,
John 14:13
Whatever you ask in my name, this I will do, that the Father may be glorified in the Son.
Acts 4:30
While you stretch out your hand to heal, and signs and wonders are performed through the name of your holy servant Jesus.”
John 14:26
But the Helper, the Holy Spirit, whom the Father will send in my name, he will teach you all things and bring to your remembrance all that I have said to you.
The Father, Son and Holy Spirit are three persons but One God.