He can't be serious. There are numerous interpretations for "kingdom of heaven" (or, Kingdom of God), one of which is, simply, "kingdom of heaven". Barnes simply cherry-picks an alternate notion that "kingdom of heaven" refers to the Church on earth, and then calls it "doubtless"? The Kingdom of Heaven is the heavenly domain of God, as well as the Church on earth, as well as Jesus Christ Himself. All three interpretations apply, but Barnes picks the one that supports his opposition to the papacy. This is part and parcel of the evangelical mindset which can only compartmentalize Scripture as an instrument to defend a particular view, instead of embracing the entirety of Scripture as an instrument of Revelation.
When he says, therefore, he will give him the keys of the kingdom of heaven, he means that he will make him the instrument of opening the door of faith to the world--the first to preach the gospel to both Jews and Gentiles.
Absolutely - at one level, this is true. But the interpretation of "kingdom of heaven", as I pointed out above, is represented in a number of equally appropriate ways - the most commonly accepted being, "the heavenly realm" itself.
The "power of the keys" was given to Peter alone solely for this reason;
Fact not in evidence. Author's opinion presented as fact.
The way I understand the terms being used by Jesus “kingdom of heaven” and “kingdom of God” is to refer to His Church. He constantly talks about it being near, or having come upon us. And his parables used to describe it (See Matthew 13 and others) always talks in terms of it being established and growing. The Apostles in their epistles reflect this same understanding (see 1Corinthians 15:24 and others). I don’t think Barnes’ interpretation is a great stretch.