The answer to the first question is no. Take for $X$ an Enriques surface; it admits an elliptic fibration $p:X\rightarrow \mathbb{P}^1$ with two double fibers, i.e. elliptic cuves $E_i$ such that $\mathcal{O}_X(2E_i)\cong \mathcal{O}_{\mathbb{P}^1}(1)$. Then one sees easily that $h^0(\mathcal{O}_X(2kE_i))=h^0(\mathcal{O}_X((2k+1)E_i)=h^0(p^*\mathcal{O}_{\mathbb{P}^1}(k))=k+1$ for all $k\geq 0$.

However I must say I have no counterexample if you ask only for $n\mapsto h^0(nD)$ to be non-decreasing for $n>>0$.

(edited) Indeed Lazarsfeld's example mentioned by user132885 answers your first question: take $X=Y\times Z$ and $\mathcal{O}_X(D)=L\boxtimes M$, with $L$ ample and $M$ torsion, say of order $p$. Then $h^0(mD)=h^0(L^m)$ if $m$ is a multiple of $p$, and $0$ otherwise. The Iitaka dimension of $D$ is $\dim Y$.