Galois representations and normal bases

Indeed the linear action of ${\rm Gal}(L/K)$ on $L$ as a $K$-vector space is the regular representation, and this does follow from the normal basis theorem: $K[G]\cong L$ via $u\mapsto u\alpha$ where $\alpha$'s $G$-orbit is a normal basis for $L/K$. This is an isomorphism of left $K[G]$-modules, not algebras, obviously.

The isomorphism type and set of possible decompositions of a representation is not dependent on any choice of vector space basis. Over algebraically closed fields $K$, the group algebra of a finite group $G$ (with order coprime to ${\rm char}\,K$) satisfies $K[G]\cong \bigoplus V\otimes_K V^*$ as both $K$-algebras and as $K[G]$-bimodules, where $V$ varies over all irreducible representations (up to isomorphism) and $V^*$ stands for the dual. This is the Wedderburn decomposition of $K[G]$.

For more general $K$ the algebra situation is not so nice - the group algebra decomposes as a direct sum of matrix algebras over division rings. Using character theory we can write down a relatively explicit $K$-algebra decomposition via primitive central idempotents:

$$K[G]\cong \bigoplus_{\chi\in{\rm \,Irr}}K[G]e(\chi), \qquad e(\chi)=\frac{\chi(1)}{|G|}\sum_{g\in G}\chi(g^{-1})g \tag{$*$}$$

This is discussed further in the blog post Idempotents and Character theory. Note that this applies to generic $G$ (with aforementioned characteristic caveat); assuming $G$ is a Galois group over $K$ probably doesn't change anything (in fact depending on answers to open inverse Galois problems, there may not be any extra utility at all to describing $G$ as a Galois group).

As $L\cong K[G]$ as a $G$-module, $G$-invariant subspaces of $K[G]$ correspond (under application to a normal basis generator $\lambda\in L$) to $G$-invariant subspaces of $L$. Maschke's theorem says that $K[G]$ is semisimple, which in particular means that every submodule of $K[G]$ is formed by adding some subset of the summands in the Wedderburn decomposition $(*)$. By passing between the algebra $K[G]$ and field $L$ via $\lambda$s we have classified all $G$-invariant vector subspaces.

As I mentioned before with Galoisness of groups having unknown relevance to the problem; the entirety more or less of what's divulged above is pure representation theory with no number theory content at all. For more information regarding the basics of representation theory there are many good resources online that are easy to find, for example Murnaghan or Etingof et al.

This doesn't mean there is never any special number-theoretic component to representation theoretic attacks on algebraic structure in number theory. If ${\cal O}_L$ and ${\cal O}_K$ are the rings of integers of $L$ and $K$ then the Galois action descends to ${\cal O}_L$ and ${\cal O}_K$ is invariant. Thus, ${\cal O}_L$ becomes a module over ${\cal O}_K[G]$ and we can ask about its structure as such. As it happens, ${\cal O}_K[G]$ is actually "too small" (not enough scalars acting on ${\cal O}_L$) to be nice; one passes to the associated order defined as ${\frak A}_{L/K}:=\{u\in K[G]:u{\cal O}_L\subseteq{\cal O}_L\}$ and one may instead consider the more natural so-called Galois-module structure of ${\cal O}_L$ over ${\frak A}_{L/K}$. Thomas surveys local field extensions.

Edit: What I originally wrote in response to the third question applies to the existence of primitive elements, not normal bases.