How does Anselm respond to the idea that existence is not a property? When Gaunilo used the perfect island as an objection to something existing in understanding it must exist in reality as well, how did Anselm respond to this objection? Please refer to Anselm's ontological arguments.

What is important to keep in mind when reading the philosophy of Anselm is that he was primarily arguing in favor of the existence of God, and any references to islands or imaginary things that exist in the understanding are really just substitutes for justifying the objective existence of God. In chapter II of Anslem’s Proslogion, he makes his most famous philosophic remark: “Now we believe you [God] to be something greater than which we can conceive of nothing.”
This is a notoriously complicated proposition, even for experts in early philosophy, but it can be interpreted somewhat like this: if God is thought of, then God must exist, because God’s very existence is something greater and more real than a human being’s conception of him. Therefore, the mere presence of God in the understanding is proof of his existence, because God’s being is greater than the understanding itself.
About a hundred years after the death of Anselm, Thomas Aquinas actually made a very similar argument (in much clearer prose) in his Summa Theologica. In the third article, Aquinas argues that the existence of God can be proven based on the logic of efficient causes. Aquinas made the argument that, because no thing can be the cause of itself, it must be necessary for something to exist that gave rise to the “first” efficient cause—something in fact very similar to Aristotle’s “unmoved mover.” Aquinas maintained that this first cause was God, and in this way his argument hearkens back to what Anselm is saying: God’s existence is greater than and lies outside of causality, because he is the one that gives existence to “cause” in the first place. Therefore, God must be greater than the understanding—as Anselm is describing it—because it is through God that understanding comes into being in the first place. Therefore, if God exists in the understanding, he exists in reality. This is how you can break down Anselm’s ontological arguments.
As for his debate with Gaunilo, the island is a metaphor for God. In their debate, the form of the argument is less important than its substance. Gaunilo could be talking about anything, but what is important is that the logic of Anselm’s argument demonstrates that existence in the understanding equates to existence in the real world. His response to Gaunilo a bit later demonstrates this principle:

It now seems obvious that a thing such that none greater can be conceived cannot be thought of as nonexistent since it exists on such firm grounds of truth. For otherwise it would not exist at all.
. . . But if he [a person who thinks of something’s existence] does think, then he thinks of something which cannot be thought of as not existing. For if it could be conceived of as nonexistent, it could be conceived as having a beginning and an end. Now this is impossible.”

It is impossible because its beginning and end would imply its objective reality as a creation of God and would therefore have to exist in the real world.
This is all a very convoluted way of thinking. When interpreting Anselm, just simplify it like this: if something is thought, then it must exist in reality as well. In other words, nothing that exists in the mind does not exist in the real world. This is because, in order for a person to manifest an image in their mind in the first place, the basic principles of this thing would have had to come from somewhere outside of the person themselves, and Anselm argues that this origin is God, who is, to repeat Anselm’s famous proclamation, “something greater than which we can conceive of nothing.”