If you read closely what they write on page 50, you will notice that they make four requirements on the action of $G$ on $P$:
(0) You are given [ahead of time, as a part of a principal fiber bundle data] a smooth manifold $M$. [This part precedes their definition.]
(1) You are given a smooth free action $G\times P\to P$ of a Lie group $G$ on a smooth manifold $P$.
(2) The quotient space $P/G$, equipped with quotient topology [they did not specify this but this is the default assumption when dealing with quotient spaces of topological spaces] is homeomorphic to $M$. [Note that at this point it is meaningless to say "diffeomorphic" since $P/G$ is just a topological space.]
(3) The projection map $f: P\to P/G\to M$ [where the latter map is the homeomorphism in (2)] is locally a product in the smooth category, i.e. for every $x\in M$ there is a neighborhood $U$ of $x$ in $M$ such that $f^{-1}(U)$ is $G$-equivariantly diffeomorphic to $G\times U$ where $G$ acts on itself via left multiplication and acts trivially on $U$.
I will refer to this as the KN-definition (of a principal fiber bundle).
This is a definition and you do not need to prove anything about it, so the answer to your question 1 is negative; the answer to your question 2 is that a smooth structure on $P/G$ is given by the requirements 0 and 2; the answer to your question 3 is also negative. In particular, the QMT (or the "slice theorem") is never relevant.
What I will prove now is that one can modify their definition so that the connection to the QMT/slice theorem is apparent.
Lemma A. KN definition implies that the action of $G$ on $P$ is proper.
This lemma will be a corollary of a stronger statement where we weaken the KN definition and still get properness of the action:
Lemma B. Suppose that $P$ is a Hausdorff locally compact topological space, $G\times P\to P$ is a continuous action of a topological group such that the quotient space $B=P/G$ is Hausdorff and the quotient map $q: P\to B$ is locally trivial in the topological sense: For every $x\in B$ there is a neighborhood $U$ of $x$ in $B$ such that $q^{-1}(U)$ is $G$-equivariantly diffeomorphic to $G\times U$ where $G$ acts on itself via left multiplication and acts trivially on $U$. Then the action of $G$ on $P$ is proper. [Note that we have no manifold assumptions anywhere in this theorem, in particular, $G$ is not assumed to be a Lie group.]
Proof. Since $P$ is locally compact, it suffices to check that any two distinct points $x, y\in P$ have neighborhoods $U_x, U_y$ in $P$ such that
$$
G_{U_x,U_y}:= \{g\in G: gU_x\cap U_y\ne \emptyset\}
$$
is relatively compact in $G$.
Consider first the case when $q(x)\ne q(y)$. Since $B$ is Hausdorff and $q(x)\ne q(y)$, these points of $B$ have disjoint neighborhoods $V_x, V_y$. Therefore, $U_x:= q^{-1}(V_x)$, $U_y:= q^{-1}(V_y)$ are also disjoint. Since $G U_x=U_x$, $G U_y=U_y$,
$$
G_{U_x,U_y}= \emptyset.
$$
Suppose now that $q(x)=q(y)$, i.e. $Gx=Gy$. Thus, without loss of generality, we may assume that $x=y$. Local triviality of the projection
$$
q: P\to B
$$
means that there exists a neighborhood $V$ of $q(x)$ in $B$ such that $q^{-1}(V)$ is $G$-equivariantly homeomorphic to the product $G\times V$ (via a homeomorphism $f$). Then $f(x)= (g,q(x))$ for some $g\in G$. Since $G$ acts on itself properly by left multiplication (I will leave this as an exercise), we can take as the required neighborhood of $x$ the subset $f^{-1}(K\times V)$, where $K$ is a relatively compact neighborhood of $g\in G$. qed
It is clear that Lemma B implies Lemma A since manifolds are Hausdorff (at least, according to the standard definition).
Now, let us connect this discussion to the QMT and slice theorem proven in
R. Palais, On the existence of slices for actions of non-compact Lie groups. Ann. of Math. (2) 73 (1961) 295-323
Proposition C. Suppose that $P$ is a smooth manifold, $G\times P\to P$ is a smooth free action of a Lie group which satisfies the assumptions of Lemma B (i.e. the quotient $B=P/G$ is Hausdorff and the projection $q: P\to B$ is locally trivial). Then $B$ has structure of a smooth manifold $M$ such that the quotient map $P\to B=M$ is a principal fiber bundle in the KN sense. Moreover such a smooth structure on $M$ is unique. [Answering the question that you asked in a comment.]
Proof. By Lemma B, the action of $G$ on $P$ is proper. Therefore, the QMT (or slice theorem) applies and, therefore, every $G$-orbit $Gy\subset P$ has a $G$-invariant neighborhood $V\subset P$ which is $G$-equivariantly diffeomorphic to the product $G\times U$, where $U$ is an open subset in $R^n$ and $n=dim(P)-dim(G)$. Equivariance of this diffeomorphism yields a homeomorphism $h: V/G\to U$. Thus, we obtain an atlas on $B$ given by the maps $h: V/G\to U\subset R^n$. The fact that transition maps are smooth follows comes from the fact that the equivariant maps $\tilde{h}: V\to G\times U$ were diffeomorphisms. Thus, $B$ is now equipped with the structure of a smooth manifold $M$. (Hausdorfness of $B$ was an assumption and 2nd countability of $B$ follows from 2nd countability of $P$.)
The projection map $q: P\to B=M$ is smooth by the construction. Since the maps $\tilde{h}$ are diffeomorphisms, we also obtain that the local trivializations appearing in Part 3 of KN-definition are diffeomorphisms. This proves the existence part of the proposition.
Let us prove the uniqueness part. Suppose that $W\subset B$ is an open subset, $h: W\to U\subset R^n, h': W\to U'\subset R^n$ are charts of two smooth atlases on $B$ satisfying the KN-requirements, i.e. there exist $G$-equivariant diffeomorphisms
$$
\tilde{h}: q^{-1}(W)\to G\times U, \tilde{h}': q^{-1}(W)\to G\times U'
$$
which project to the maps $h, h'$. I claim that the transition maps $h'\circ h^{-1}$ is smooth, hence, atlases are equivalent. Indeed, the composition
$$
\tilde{h}'\circ \tilde{h}^{-1}: G\times U\to G\times U'
$$
is smooth (since $\tilde{h}, \tilde{h}'$ are diffeomorphisms) and, due to its $G$-equivariance, has the form
$$
(g,u)\mapsto (\phi(g), h'\circ h^{-1}(u)).
$$
This implies smoothness of $h'\circ h^{-1}$. qed
One last thing, I initially thought that in Lemma B it suffices to assume that the quotient space $B$ is Hausdorff to conclude properness of the action (i.e. that the local triviality part is redundant). I now realized that I was wrong:
Example. There is an example (derived from the Reeb foliation) of a smooth free action of $G={\mathbb R}$ on $P={\mathbb R}^2- \{(0,0)\}$ such that the quotient space $P/G$ is homeomorphic to $[0,\infty)$ (and, hence, is Hausdorff) but the action of $G$ on $P$ is not proper.
I do not know if there are similar examples (a nonproper smooth free action of a Lie group on a smooth manifold, $G\times X\to X$) where $X/G$ is a topological manifold (without boundary). Here is what I do know:
Theorem. Suppose that $G$ is a Lie group with finitely many connected components acting smoothly and freely on a smooth manifold $X$ such that $B=X/G$ is a topological manifold $M$ which admits a smooth structure such that the quotient map $X\to M$ is a submersion. Then the action of $G$ on $X$ is proper and, hence, $G\times X\to B$ is a principal fiber bundle.
