$\DeclareMathOperator\Fr{Fr}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Sym{Sym}$Another, more geometric, approach to this question is as follows: let $G$ be a Lie group acting on $M$, and let $P$ be a principal $H$-bundle over $M$. (This addresses your question (2), but it includes (1) if you take $P = \Fr(V)$ to be the frame bundle of the vector bundle $V$.)
Now, if the action of $g \in G$ is supposed to lift to $P$, the least we need is an isomorphism $\psi_g \colon P \to \Phi_g^*P$ of $H$-bundles over $M$. For these to actually provide an action, they must satisfy that $\Phi_g^*\psi_h \circ \psi_g$ agrees with $\psi_{hg}$ under the canonical isomorphism $\Phi_g^*\Phi_h^*P \cong (\Phi_h \circ \Phi_g)^*P \cong \Phi_{hg}^*P$.

This can be rephrased more globally as follows: let $\Aut(P)$ denote the group consisting of all diffeomorphisms $P \to P$ over $M$ (i.e. mapping fibres to fibres) that preserve the $H$-action. That is, these are bundle maps which cover any diffeomorphism $M \to M$. This comes with a projection $p \colon \Aut(P) \to \Diff(M)$. The map $p$ is a group homomorphism, and if we consider only diffeomorphisms isotopic to the identity on $M$, then it is even surjective (you can see this by a cohomology argument, or by endowing $P$ with a connection).
Let $\mathcal{G}(P)$ be the fibre of $p$ over $1_M$ (this is the group of gauge transformations of $P$). We thus have a short exact sequence
$$\mathcal{G}(P) \to \Aut(P) \to \Diff_0(M)\,,$$
where $\Diff_0(M)$ denotes the connected component of $1_M$ in $\Diff(M)$.
This is even a short exact sequence of smooth groups (for instance in the world of diffeological spaces).

Our $G$-action is a smooth group homomorphism $\Phi \colon G \to \Diff(M)$, and I will assume that it factors through $\Diff_0(M)$. (Alternatively, assume that for every $g \in G$ a $\psi_g$ as above exists; otherwise there is no chance of finding a lift of the $G$-action $\Phi$ to $P$ anyway.)
Then, we obtain a pullback sequence
$$\mathcal{G}(P) \to \Sym(P) \to G\,,$$
where $\Sym(P) = \Phi^*\Aut(P)$.

A lifting of the $G$ action on $M$ to one on $P$ is now the same as a splitting of the above short exact sequence of *smooth* groups. The obstructions to the existence of lifts thus are the obstructions to splitting this sequence smoothly (I am not sure how well non-abelian group cohomology for diffeological groups is developed, but I would expect it to describe the obstructions; if $G$ is discrete, ordinary group cohomology should suffice). If you find lifts, then the sequence shows that these lifts are a torsor over $\mathcal{G}(P)$; this controls how many lifts there are in case there are any. You can also find a bit more about this in Bunk, Müller, and Szabo - Smooth 2-Group Extensions and Symmetries of Bundle Gerbes.