ID is looking for examples of traits that cannot be explained naturally. Archaeologists have to do this all the time, so it's not impossible. If they find evidence of a fire in a cave, for example, they need to determine if it's a manmade fire pit or simply some sort of natural fire. To make that evaluation, they'll look at the features of the fire for evidence of design. ID seeks to do the same with biological features. Specifically, it is looking for biological traits that cannot be adequately explained through a process of random mutations and natural selection.
And, yes, sometimes in archaeology, the answer is, "We don't know." But unlike your assessment of science, they don't automatically give the unknown over to a natural process. Now you could argue that science defaults to a natural explanation because there is no proof of God, unlike the fact that we know that human beings exist and could have made the fire if they were there. To that I'll simply say that for many religious people, they are convinced that God exists and is real in a variety of ways, thus they don't automatically assume a natural explanation.
And what assumptions? Can you name anything in the history of science where the assumption of supernatural or extranatural causes has been required?
Not off hand, though there are certainly explanations provided by science that rely as much on the assumption that a natural explanation exists as actual evidence that a natural explanation does exist. In other words, there are areas where science accepts the mere possibility of a natural explanation to be sufficient to exclude any other explanation. And if you want to be anal-retentive about the scientific method, you should realize that such hand-waving doesn't really prove anything. It's just begging the question.
I also want to point out that (A) absence of evidence is not evidence of an absence and (B) just because something works 99 times in a row does not mean it will work the same way on the 100th time. Newtonian physics seem to work just fine until you start looking at high energy physics and different frames of reference. You won't notice that Newtonian physics is lacking doing simple tests and just because you've done the simple tests doesn't mean Newtonian physics is correct and Relativity is a crackpot idea. In fact, they are still doing experiments trying to prove some of the more unusual effects of relativity.
Can you name any problem currently under investigation that has reached a dead end in terms of assumptions or methodology?
Off hand? No. But there are plenty of places where science is currently waving it's hand and making assumptions with insufficient supporting evidence. Please note that I'm not claiming that science should be abandoned or that natural explanations for biological features shouldn't be explored. What I am saying is that thinking you have all the answers and confusing evidence and assumptions may not give you the correct answer and, as I've mentioned, there are plenty of examples in the history of science where the establishment resisted better theories because they thought they knew it all.
I'm going to grant what I think is one benefit of ID. It has required mainstream science to tighten up its terminology, and is currently forcing popular publications to be more careful in their pronouncements. It might, in the long run, result in the demise of some of the crap known as deconstructionism. All these things are good.
Correct. And it might also encourage funding for researchers to try to understand if and how various biological systems work and evolved rather than assuming that they just did.
That is the work of science, to find explanations.