What is a Theory?

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Popper implies that a falsifiable theory will eventually be falsified.

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As I have stated elsewhere this is to be expected, based on the incomplete state of our knowledge. It is not inherent in being falsifiable (indeed any true and useful theory must be falsifiable but cannot be falsified).

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If every falsifiable theory will eventually be falsified, it means that every falsifiable theory will eventually be proven to be false. That means that the correct status of scientific theories is: unproven, false.

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This is important. Too many people use the phrase "scientifically proven", even though, not one single theory has ever been "scientifically proven".

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No. What must be realised is that when we talk of scientific proof we are not speaking of the absolute proof provided by inductive logic. We are, however, speaking of sufficiently strong supporting evidence that it would be sensible to accept the theory as true (as indeed, we can be sure that it IS mostly true).

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The value of knowledge does not reside in its truth, but in the fact that it is very hard to prove that it is false.

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That is just hideously wrong. Especially in science, which we are discussing. A theory that makes useful and accurate predictions is valuable. A theory that does not is of relatively little value.Pointless speculation is not valuable knowledge, no matter how difficult it would be to disprove.

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It is not really philosophy. Given the stated definitions of proven/unproven and true/false, science is, in its own terms, unproven and false.

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I'll agree that it's not philosophy. All you are doing is redefining "truth" so that you can say that all unfaalsifiable statements - even the false ones - are "true".Unfortunately for you, you can't even manage to get that simple piece of trickery right. By your definition a theory isn't false UNTIL the falsifying obervation has been made.

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The reason why I am interested in the limitations of science, is because science is often used to attack religion.

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In other words you are really interested in redefining words to dishonestly try to make religion look good. Which only succeeds in amking religion look bad.

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But then again, the underlying (but unproven) assumption is that an infinitely falsifiable theory, such as "Water boils at 100 C", must be false, even if we have not made the falsifying observation as yet.

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No. By your definition it is NOT false until the falsifying observation is made. And by the normal definitions a true theory would be "infinitely falsifiable" and thus our argument is still wrong whether we use your rigged definitions or not. (And nobody with any sense would use your rigged definitions since thier only purpose is to deceive).