The gospel of Jesus Christ: What is it?

[Arkady]

Did receiving the Spirit have nothing to do with being given the keys to the kingdom? If so, then when exactly did Jesus give His disciples that key?

Also, what is the "key?" Is it not the Spirit?

Yes. It is the Spirit.

Words of Symeon the New Theologian > Volume 1 > Chapter 3 Three Hypostases of The Godhead are equal in everything (from russian translation)
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The door is the Son: "I am the door" (John 10:9). The keys to the door are the Holy Spirit: "Receive ye the Holy Ghost: Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained." (John 20:22-23). The Father is the house: "In my Father's house are many mansions" (John 14:2).

Tractates on the Gospel of John (Augustine) > Tractate 124
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The Church, therefore, which is founded in Christ received from Him the keys of the kingdom of heaven in the person of Peter, that is to say, the power of binding and loosing sins.

Concerning Repentance (Ambrose) > Book I Chapter 7
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32. And You indeed, O Lord, desirest that all should be healed, but all do not wish to be healed. Novatian wishes not, who thinks that he is whole. You, O Lord, sayest that You are sick, and feelest our infirmity in the least of us, saying: I was sick and you visited Me. Matthew 25:36 Novatian does not visit that least one in whom You desire to be visited. You said to Peter when he excused himself from having his feet washed by You: If I wash not your feet, you will have no part with Me. John 13:8 What fellowship, then, can they have with You, who receive not the keys of the kingdom of heaven, saying that they ought not to remit sins?

On the Holy Spirit (Ambrose) > Book III Chapter 17
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123. The law of Your mouth, O Lord, is good unto me, I keep Your commandments. You have Yourself said that You are one with the Father. Because Peter believed this, he received the keys of the kingdom of heaven, and without anxiety for himself forgave sins. Judas, because he believed not this, strangled himself with the cord of his own wickedness.