Intuitively, I guess all the statements are true so there should exist examples for each. But I can't think of specific ones + not 100% sure of my intuition.
Can you guys provide solid examples with fully-structured steps?
** The space is strictly limited to the Euclidean one, and I need specific theorems/definitions used to validate the answers if possible, i.e., Weierstrass M-Test, properties of Taylor Series, definition of a power series, and etc.
Let me provide you with an (probably unsatisfying) answer:
(a) Take $g(x)=\frac{1}{1+x^2}$. This function is as good as a function on $\mathbb R$ can be, i.e. infinitely often differentiable. Why does its Taylor series only have radius of convergence $R=1$? That's due to the fact that this function is the restriction of the function $f:\mathbb C \to \mathbb C, f(z)=\frac{1}{1+z^2}$ which has a pole in the points $z=\pm i$ due to the fact that $i^2=-1$. But you don't need complex analysis to see this. Just calculate the radius of convergence by one of the usual formulas.
(b) Due to the very helpful comments by Jyrki, here is an example to part (b): $h(x)=\sin(x)+e(x)$, where $e(x)=\exp(-\frac{1}{x^2})$ for $x\neq 0$ and $e(0)=0$. $e(x)$ is in fact infinitely often differentiable and has the same Taylor series as $\sin(x)$, although it equals $\sin(x)$ only for $x=0$.
(c) As I learned from the link commented by Sean, (c) indeed is true.
Before voting down this answer, let me know where I can improve it ;-) Comments are welcome.
EDIT: I tidied up the answer since the OP seemed to be confused by parts of it.