Well, alien illness probably won't kill us. Without a common ancestry, it's highly unlikely that anything that uses an alien as a host can live in our bodies.
We don't know that. There are plenty of examples on Earth of cross-species infections. No telling how microbial/fungal agents could directly attack or indirectly impact the delicate balance here on Earth.
Further, any microbe hitching a ride aboard an alien may well find Earth's general environment to be easy to thrive in. Think Kudzu, but at the microbial level. Eventually, the alien microbe and/or the natural Earth microbes will mutate in order to out-survive the other.
You assume that all life would be carbon based. And even assuming a "super fit" carbon-base alien microbe, the worst you could get would be some kind of eco-disaster that will end up being controlled anyway - like some sort of alien-algae overpopulating our oceans doing some eco-havoc. Even that would involve a kind of "super-fitness" created by an accelerated evolution running for much longer than here on Earth, as this microbe will have to compete with many others that are much more adapted to our environment.
Anything really dangerous, like something that can infect multiple species, would have to be "engineered". "Mother nature" (aka evolution) doesn't engineer bio-weapons made to take over alien ecosystems - you would need huge time spans plus ubiquitous means of interstellar transportation for natural selection to favor such features (think "star gates" but zillions of them with tons of creatures and cargo running through them for at least tens of millions of years).
So I'd take contact with a "non-engineered super infection" like you describe as a sign that someone has been operating a huge network of interstellar transportation systems for quite some time, as this would provide the only "peacefull" evolutionary history (again, excluding civilizations expert in bio-war things...) for such thing to evolve, so it would be quite a good omen for hopes "intergalactic fraternity" :)
The possibility of non-carbon based life is so remote as to be irrelevant. We'd have to go through many star birth and supernovae cycles to get to the point where silicon is even anywhere near as abundant as carbon is now.
As for microbes from other worlds, who knows. The properties of water are pretty much required for life (as far as we know) and so they'd have 0 trouble interfacing with most organisms. Combine that with the fact that the basic amino acids can form in a wide range of conditions and you've got the recipe for interference. Also, infections happen when organisms invade and multiple, some times the damage is a secondary side effect (i.e. waste or toxic byproducts)
