"skyscraper group scheme"

Yes: the following examplpe is quite different from the example by Ariyan, although the generic and closed fibers are the same as his.

Start with the constant group scheme $A:=(\mathbb{Z}/2\mathbb{Z})_S$, and consider the closed subscheme which is the union of the zero section and the closed point of the other section. This is immediately seen to be a subgroup scheme of $A$, with generic fiber $0$ and closed fiber $\mathbb{Z}/2\mathbb{Z}$. It is finite over $S$ (in particular separated) but of course not flat.

You can construct infinitely many variants by replacing the closed point by any infinitesimal neighborhood of it.

If you take a stable curve $C$ over $S$, it it known that its automorphis functor $\mathscr{G}:={\underline{\mathrm{Aut}}}(C/S)$ is a finite unramified $S$-group scheme, hence it "looks like" the above example, rather than Ariyan's nonseparated one. In particular ($K$ being the fraction field), the restriction homomorphism $\mathscr{G}(S)\to \mathscr{G}(K)$ is bijective, so if $C_K$ has no $K$-automorphisms then if $C$ has no $S$-automorphisms.

Yes.

Let $\mathbb{A}^{1,2}$ be the affine line with a double origin. Consider the natural map $\mathbb{A}^{1,2}\to \mathbb{A}^1$. (To define this map, let $0_1$ and $0_2$ be the origins in $\mathbb{A}^{1,2}$. The above map sends any $x\neq 0_1, 0_2$ to $x$. It sends $0_1$ and $0_2$ to the origin in $\mathbb{A}^1$.)

The above morphism realizes $\mathbb{A}^{1,2}$ as a quasi-finite flat group scheme over $\mathbb{A}^1$. (It is not a separated group scheme over $\mathbb{A}^1$, of course.)

Note that the generic fibre of this group scheme is the trivial group. The fibre over the origin is the group $\mathbb{Z}/2\mathbb{Z}$.

To get to the situation you desire, let $\mathrm{Spec} \mathbb{C}[[t]]\to \mathbb{A}^{1}$ be a dominant map whose image contains the origin. Now base-change the group scheme $\mathbb{A}^{1,2}\to \mathbb{A}^1$ along this morphism to get a group scheme $$ G\to \mathrm{Spec} \mathbb{C}[[t]]$$ with trivial generic fibre and a non-trivial special fibre. (Here $G=\mathbb{A}^{1,2}\times_{\mathbb{A}^1} \mathrm{Spec} \mathbb{C}[[t]]$.)