As an answer to (1), consider any compact subgroup $G\subset O(n)$ and look at the $G$-invariant functions on $\mathbb{R}^n$ that are $G$-invariant and have homogeneity $1$. As long as $G$ does not act transitively on the space of lines in $\mathbb{R}^n$, that set of functions will be properly larger than the $O(n)$-invariant functions of homogeneity $1$ 1$, and any one of them that is sufficiently near the standard norm will be a norm.
As a specific example, consider the irreducible action of $SO(3)$ on the $5$-dimensional vector space $S$ of traceless symmetric $3$-by-$3$ matrices. There are two $SO(3)$-invariant polynomials on $S$, S$: $p_2(m) = trace(m^2)$ and $p_3(m) = det(m)$, and they . (They generate the ring of $SO(3)$-invariant polynomials on $S$. S$.) Now let $$ |m| = \bigl(p_2(m)^3 + \epsilon\ p_3(m)^2\bigr)^{1/6}. $$ for some small $\epsilon$. When $\epsilon>0$ is sufficiently small, this will be a strictly convex norm on $S$, and its symmetry group will be $SO(3)$, which acts irreducibly on $S$.
