I came across those questions on my school's topological dynamical-system textbook and I had totally no idea about them.
(a) There exist a compact metric space $X$, a homeomorphism $f:X\rightarrow X$ and a point $x_0\in X$, such that $$\text{Orb}_f(x_0)={\{f^k(x_0):k\in\mathbb{Z}\}}$$ is dense in $X$ while ${\{f^k(x):k\in\mathbb{Z}^+\}}$ and ${\{f^k(x):k\in\mathbb{Z}^-\}}$ are not dense in $X$ for any $x\in X$.
(b) There exist a metric space $X$ with countable many open dense subset $\{U_i\}$, s.t. $\bigcap_{i}U_i=\varnothing$.
Thanks to Henno Brandsma, now we have (b):
Consider $X=\mathbb{Q}(=\{r_i\}_{-\infty}^{+\infty})$ as the subspace of $\mathbb{E}^1$ and $U_i=X-\{r_i\}$. Clearly, every $U_i$ is open since $U^c_i=\{r_i\}$ is a point set which is closed. To see the density of $U_i$, it is sufficient to show that $U_i$ is not closed. In fact, sequence $\{r_i+\frac{1}{n}\}\subset X$ converges to $\{r_i\}\notin U_i$, which implies that $U_i$ is not closed.
EDITED! Sorry I've given the (a) incorrectly and now it is clarified. The first version is wrong because that $\bigcup_{x\in X}{\{f^k(x):k\in\mathbb{Z}^-\}}=X$ and $\bigcup_{x\in X}\text{Orb}_f(x)=X$ always holds.
Additional Question:
(b)$^\prime$ Determine whether it is possible that there exist a metric space $X$ with countable many open dense subset $\{U_i\}$, such that for any $i\neq j$, $U_i\cap U_j=\varnothing$ holds.
Just now I figured out (b)$^\prime$:
If for any $i\neq j$, $U_i\cap U_j=\varnothing$, we will have $U_i\subset X-U_j$. Then $\overline{U_i}\subset X-U_j$ because $X-U_j$ is closed. Clearly $X-U_j$ is a proper subset of $X$, which implies $\overline{U_i}\neq X$.
Let $X = \{-1, 1 \} \cup \{ \frac{n}{1+ \lvert n \rvert} : n \in \mathbb Z \}$ with the subspace topology inherited from $\mathbb R$. Let $x_n = \frac{n}{1+\lvert n \rvert}$. Define $f : X \to X, f(-1) = -1, f(1) = 1, f(x_n) = x_{n+1}$ for $n \in \mathbb Z$. This is a homeomorphism. You have $f^k(x_0) = x_k$. Thus $$\text{Orb}_f(x_0) = \{ \frac{n}{1+\lvert n \rvert} : n \in \mathbb Z \}$$ which is dense in $X$ and $${\{f^k(x):k\in\mathbb{Z}^\pm\}} = \{ \frac{n}{1+\lvert n \rvert} : n \in \mathbb Z^\pm \}$$ which are not dense in $X$.