Can somebody help me with some very basic questions about the fundamentals of information theory?
What is the nature of this relationship? There are many possibilities. Shannon used a logarithm because he wanted information to be additive, that is, $I(xy)=I(x)+I(y)$ for independent events $x$ and $y$.

You're correct that the arrival of a message at the receiver decreases its uncertainty. It may not be decreased to zero, because there may be an error in the communication. Also, a given channel can only support a certain information rate, which is its capacity.
It does not make sense to ask about messages of probability zero. The source alphabet is defined so that all symbols have probability greater than zero.