Well, the "virus is alive" thing is more of a toy example, a real example (which actually happened) that pertains to biology might be that of cigarettes causing lung cancer. There were extensive studies done to prove the statistical link between smoking and lung cancer but in the end we do not find a one to one correlation. Some smokers will go their entire lives and not get cancer, other people will never smoke and get lung cancer (although that number is very small). It can be obviously established that smoking is statistically linked to getting cancer, but the burden for claiming that smoking "causes" lung cancer is somewhat higher. Since scientists at the time could not account for why some smokers develop lung cancer and others do not it becomes question of epistemology, the branch of Philosophy that deals with knowledge. How can we really know that lung cancer "causes" cancer or if it is simply a side effect of something intermediate caused by smoking or closely associated with it. The other alternative is that smoking is like playing roulette. In that case it also seems inappropriate to say that smoking "causes" cancer and instead it causes one to have an increased risk. (I don't know if more research has been done on why some people develop cancer and others don't.) Causation is an extremely thorny subject but in this case (iirc) it was one of the first times that a phenomenon was accepted as being a cause when there was not a one to one direct correlation with its effect.

As for the tree in the woods, again this is a toy example. It's a vastly simplified example that philosophers use to discuss epistemology because it provides a simple basis on which to argue about something. In math and engineering they use the spherical cow in a friction-less world. Obviously no one cares about a spherical cow but it's useful as an isolation tool so that you can really work on just the problem at hand.

I don't know a lot about string theory but if it cannot be tested and is still considered science this is indeed very troublesome for many theories of science. Perhaps this ties into the realism / anti-realism debate where philosophers debate whether there are actually strings or they are just convenient mathematical constructs with wide explanatory power. I personally am somewhat of a nihilist on this point and I don't think it's appropriate to bend the concept of reality to apply to things like strings and that this is basically just an incoherent exercise to begin with.

A non-string example of this debate that I know a little bit more about (but not much) is the debate over observability. I can observe the wall in front of me unaided with my own eyes and sense of touch.

We can also "observe" radioactive decay in a gas chamber by examining the condensation trails of particles traveling through the gas, but are we really observing them? This seems to be more indirect than the first example, so can we really know that there are particles traveling through the gas? If indirect evidence is not acceptable in science, what is the cutoff point that it becomes unacceptable. I could hear from a friend of a friend of a friend that they saw bigfoot and while everyone might trust everyone else this most certainly is not a scientific observation. This latter example is extreme but it demonstrates the importance that scientists understand the scientific method and the philosophy behind it.

If string theory can only be proven indirectly, at what level of indirection does it become inappropriate to say that the observations are evidence for the theory. It sounds as if we only have very indirect evidence for string theory which is probably why it is so controversial.