It's not clear to me exactly what "$1$-dimensional dynamical system" means, so I'll stick with continuous (but not necessarily invertible) transformations of $S^1$, and with the $\omega$-limit taken in forward time.
$2$. The map $f \colon [-1,1] \to [-1,1]$ given by $f(x):=4x^3-3x$ has a dense orbit and fixes $1$ and $-1$. Take the disjoint union of two copies of $[-1,1]$ with this map defined on each. Now identify $1$ on the first interval with $1$ on the other, and identify $-1$ with the copy of $-1$ too. This creates a map of the circle with two invariant closed intervals. The "dense orbit" in each interval has $\omega$-limit set equal to the whole interval (i.e. a closed semicircle). I think that with a little care one could come up with an analytic example.
$5$. Asaf's proposed answer to 4 can be adapted to answer 5. Let $f \colon \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ be the doubling map $f(x)=2x$, and let $x=\sum_{n=1}^\infty 2^{-n^2}$. A point is in the $\omega$-limit set of $x$ if and only if it is either $0$ or $2^{-k}$ for some integer $k$.