Example of Hausdorff space $X$ s.t. $C_b(X)$ does not separate points?

Looking at the table at the back of Counterexamples in Topology, the first space which is Hausdorff but not Urysohn (which in that book means $C(X)$ doesn't separate points) is the relatively prime integer topology. That is, the topology on $\mathbb{N}\setminus \{0\}$ with basis all sets of the form $\{b+na|n \in \mathbb{N}\}$ for $a$, $b$ coprime.

Hausdorff is easy: if $k, l \in \mathbb{N}\setminus \{0\}$, take a prime $p$ larger than both and consider the open neighbourhoods $\{k+np \}, \{l+np \}$.

$C(X)$ fails to separate points because any two basic open sets have nondisjoint closures, since the closure of $\{b+na \}$ contains all multiplies of $a$.

I'll just point out that boundedness is not the obstruction: any space that admits a nonconstant real-valued continuous function admits one which is bounded, and if $C(X,\mathbb{R})$ separates points then so does $C_b(X,\mathbb{R})$. This is easy to see: if $f$ is continuous and $f(x_1)=a < b = f(x_2)$, then $g(x) = (f(x) \vee a) \wedge b$ is continuous and bounded and also has $g(x_1)=a<b=g(x_2)$.

However, continuous functions with compact support may not be enough if $X$ is not locally compact, even if it is metrizable. For instance, it's a nice exercise to check that if $X$ is an infinite-dimensional Banach space, the only real-valued continuous function on $X$ with compact support is the zero function, i.e. in your notation $C_{00}(X) = \{0\}$. (I think most authors call this space either $C_0(X)$ or $C_c(X)$.)

I recommend this short 1971 note of T.E. Gantner. He first proves:

Theorem: Each regular space $Z$ can be embedded as a subspace of a regular space $Q(Z)$ such that every continuous, real-valued function on $Q(Z)$ is constant on $Z$.

He applies the theorem as follows: take $X_0$ to be a one-point space, and inductively, for all $n \in \mathbb{N}$, define $X_{n+1} = Q(X_n)$. Let $X = \lim_{n \rightarrow \infty} X_n$, endowed with the direct limit (or final) topology. Then $X$ is an infinite regular space on which every continuous real-valued function is constant.