Entropy stays the same with a larger distribution? Based on the given definition of entropy, $H(P(X)) = -\sum_i P(x_i)log_2(P(x_i)$, it appears that if I have a distribution $P_1(x) = [\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}]$ and another distribution $P_2(x) = [\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},0,0,0,0...]$, then they have the same entropy? 
That just doesn't seem to make sense to me based on my notion of entropy. In $P_1$ we are quite uncertain whereas in $P_2$ we are relatively more certain...
 A: 
In P1 we are quite uncertain whereas in P2 we are relatively more certain...  

Obviously our issue here is not of formal treatment ($P_1(x)$ and $P_2(x)$ are formally equivalent representations of the exact same distribution -in fact, $P_2(x)$ is the more "correct" one, in a sense), but of intuition, and I will try to provide some.
The expression 
$$P_2(x) = \left\{\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},0,0,0,0...\right\}$$
includes the probabilities of the distribution, each corresponding to some value that the related random variable takes. So what you are saying is
"If I know that the random variable $X$ takes the values $\{a,b,c,d\}$ each with probability $1/4$, and also an infinite number of values each with probability zero, then I am relatively more certain compared to if I knew that $X$ takes "only" the values $\{a,b,c,d\}$ each with probability $1/4$". Why, really? Why introducing in the picture zero-probability events makes you relatively more certain? Since they were always there in the first place? 
Maybe you are thinking that $P_1(x)$ describes a situation where we don't know what other "values" of $X$ may be out there, and so we feel "uncertain", while, if we look at $P_2(x)$, we "know" that these other values have probability zero, and so we are certain that our uncertainty is contained in the four values with positive probability?
This is a very realistic description of a real-world situation -but such a real-world case would not be described by the juxtaposition of $P_1(x)$ and $P_2(x)$. Consider a new social phenomenon that can be quantified. At the beginning, do we know what values can it take? No. We will have to see it evolve through time, and still we will never be certain of all its possible values. Assume that after some time, we have observed that this phenomenon takes the values $\{a,b,c,d\}$ with practically equal probabilities. We have not observed the phenomenon acquiring any other value, not even once. Say we want to model this phenomenon, and include it in some more general model of ours. What are we to do?
As I wrote in the beginning, specifying either $P_1(x)$ or $P_2(x)$ is the exact same thing. In other words $P_1(x)$ is  not a way to express any uncertainty regarding the "possible full support of $X$" -it is just a shorthand for $P_2(x)$ that does not hurt neither the mathematics nor the inference. The very real concern that our model me be inaccurate, regarding also the support of $X$ (let alone the allocation of probabilities), is not the kind of uncertainty that is captured by the concept of Shannon's Entropy. Shannon's Entropy summarizes a fully-defined and fully described uncertainty -not uncertainty that we just know or suspect that it exists, but we are unable to tell or do anything about it.
(This is reminiscent of the old Knightian distinction in Economics between "risk"(described uncertainty) and "uncertainty"(which simply means "we are at a loss").)  
I hope all these are not irrelevant to your concerns.
