The questions about manifolds admitting flat conformal or projective structures have been answered, so I'll just address your final question, which is whether $\mathrm{PGL}(n{+}1,\mathbb{R})$ and $\mathrm{O}(n{+}1,1)/\{\pm I_{n+2}\}$ are 'maximal' finite dimensional groups acting on $n$-manifolds.

The answer depends on what you mean by 'maximal'. They are maximal in the sense that any finite dimensional Lie subgroup of $\mathrm{Diff}(\mathbb{RP}^n)$ that contains $\mathrm{PGL}(n{+}1,\mathbb{R})$ must be equal to $\mathrm{PGL}(n{+}1,\mathbb{R})$ and any finite dimensional Lie subgroup of $\mathrm{Diff}(S^n)$ that contains $\mathrm{O}(n{+}1,1)/\{\pm I_{n+2}\}$ must be equal to $\mathrm{O}(n{+}1,1)/\{\pm I_{n+2}\}$. This is essentially due to Élie Cartan, who proved the corresponding statement that the corresponding Lie algebras of vector fields are maximal finite dimensional Lie algebras in $\mathrm{Vect}(M)$ in his series of papers on so-called 'infinite groups' of diffeomorphisms. For a more modern treatment, see Singer and Sternberg's "The infinite groups of Lie and Cartan, Part I (the transitive case)".

However, if my maximal, you mean 'highest (finite) dimensional Lie subgroup of $\mathrm{Diff}(M^n)$', then this is false, as there is no upper bound for trivial reasons: Take $k$ disjoint open sets $U_j\subset M$ each with compact closure, and let $X_j$ be a nonzero vector field on $M$ that is supported in $U_j$. Then the flows of the $X_j$ commute and generate an effective action of $\mathbb{R}^k$ on $M^n$. Since $k$ can be arbitrary, it follows that there is no upper bound on the dimension of Lie subgroups of $\mathrm{Diff}(M)$.

Of course, this example is not satisfying, and you well might ask whether there is an upper bound for transitive actions. Even then, the answer is 'no'. For example, for a fixed integer $m$, consider the transformations of $\mathbb{R}^2$ that are of the form $$ \Phi(x,y) = (\ ax{+}b,\ cy + d_m x^m + d_{m-1} x^{m-1} + \cdots + d_1 x + d_0\ ) $$ where $a,b,c, d_0,\ldots, d_m$ are constants with $ac\not=0$. These constitute a Lie group of dimension $m{+}4$ acting effectively and transitively on $\mathbb{R}^2$. A similar example can obviously be constructed on $\mathbb{R}^n$ for any $n>1$. One can construct transitive examples for any $n>1$ on compact manifolds as well.

The key missing condition necessary for classification is to ask that the group act both transitively and primitively, i.e., that the group action not preserve any nontrivial foliation of $M$. Then there is a classification, due in large part to Élie Cartan, but finally completed to modern standards of rigor by Ochiai in the 70s. The conformal and projective transformation groups are indeed maximal among primitive, transitive actions, but there are many others. For example, the action of $\mathrm{PGL}(n,\mathbb{R})$ on the Grassmannian of $k$-planes in $\mathrm{R}^n$ is maximal, primitive, transitive for every pair $(k,n)$. For a given dimension $n$ of the underlying manifold, the projective group, at dimension $n^2{+}2n$, is the largest primitive transitive Lie subgroup in overall dimension.

The questions about manifolds admitting flat conformal or projective structures have been answered, so I'll just address your final question, which is whether $\mathrm{PGL}(n{+}1,\mathbb{R})$ and $\mathrm{O}(n{+}1,1)/\{\pm I_{n+2}\}$ are 'maximal' finite dimensional groups acting on $n$-manifolds.

The answer depends on what you mean by 'maximal'. They are maximal in the sense that any finite dimensional Lie subgroup of $\mathrm{Diff}(\mathbb{RP}^n)$ that contains $\mathrm{PGL}(n{+}1,\mathbb{R})$ must be equal to $\mathrm{PGL}(n{+}1,\mathbb{R})$ and any finite dimensional Lie subgroup of $\mathrm{Diff}(S^n)$ that contains $\mathrm{O}(n{+}1,1)/\{\pm I_{n+2}\}$ must be equal to $\mathrm{O}(n{+}1,1)/\{\pm I_{n+2}\}$. This is essentially due to Élie Cartan, who proved the corresponding statement that the corresponding Lie algebras of vector fields are maximal finite dimensional Lie algebras in $\mathrm{Vect}(M)$ in his series of papers on so-called 'infinite groups' of diffeomorphisms. For a more modern treatment, see Singer and Sternberg's "The infinite groups of Lie and Cartan, Part I (the transitive case)".

However, if by maximal, you mean 'highest (finite) dimensional Lie subgroup of $\mathrm{Diff}(M^n)$', then this is false, as there is no upper bound for trivial reasons: Take $k$ disjoint open sets $U_j\subset M$ each with compact closure, and let $X_j$ be a nonzero vector field on $M$ that is supported in $U_j$. Then the flows of the $X_j$ commute and generate an effective action of $\mathbb{R}^k$ on $M^n$. Since $k$ can be arbitrary, it follows that there is no upper bound on the dimension of Lie subgroups of $\mathrm{Diff}(M)$.

Of course, this example is not satisfying, and you well might ask whether there is an upper bound for transitive actions. Even then, the answer is 'no'. For example, for a fixed integer $m$, consider the transformations of $\mathbb{R}^2$ that are of the form $$ \Phi(x,y) = (\ ax{+}b,\ cy + d_m x^m + d_{m-1} x^{m-1} + \cdots + d_1 x + d_0\ ) $$ where $a,b,c, d_0,\ldots, d_m$ are constants with $ac\not=0$. These constitute a Lie group of dimension $m{+}4$ acting effectively and transitively on $\mathbb{R}^2$. A similar example can obviously be constructed on $\mathbb{R}^n$ for any $n>1$. One can construct transitive examples for any $n>1$ on compact manifolds as well.

The key missing condition necessary for classification is to ask that the group act both transitively and primitively, i.e., that the group action not preserve any nontrivial foliation of $M$. Then there is a classification, due in large part to Élie Cartan, but finally completed to modern standards of rigor by Ochiai in the 70s. The conformal and projective transformation groups are indeed maximal among primitive, transitive actions, but there are many others. For example, the action of $\mathrm{PGL}(n,\mathbb{R})$ on the Grassmannian of $k$-planes in $\mathrm{R}^n$ is maximal, primitive, transitive for every pair $(k,n)$. For a given dimension $n$ of the underlying manifold, the projective group, at dimension $n^2{+}2n$, is the largest primitive transitive Lie subgroup in overall dimension.