Möbius and projective 3-manifolds
The first examples of closed 3-manifolds not admitting a conformally flat (Mobius) structure were given by Goldman: 3-manifolds modeled on the Nil geometry (this link was given by Macbeth in the comments above). Sol manifolds also don't have a Mobius structure, whereas 3-manifolds modeled on the other six geometries do. Misha Kapovich has many results on conformally flat structures. In his thesis, he shows that certain classes of Haken 3-manifolds have finite-sheeted covers which are (uniformizable) conformally flat, and he shows there is a graph manifold which has no conformally flat structure, but which has a finite-sheeted cover which does. Kulkarni showed that connect sums of conformally flat manifolds are conformally flat. On the other hand, Kapovich's student Hwang showed that for any 3-manifold $M$, there is a 3-manifold $N$ such that $M \# N$ is conformally flat. I don't know of any recent activity on the topic.
As for your question on maximal Lie group actions on 3-manifolds, I've heard this before too, but I don't know a reference. I think you can prove it by analyzing the action of the isotropy group of a point on the jet space at that point.
Addendum: I had a discussion with Cooper about the last question (maximal Lie groups acting faithfully on an $n$-manifold), and we have an idea how to approach it (at least for smooth actions). If a Lie group $G$ acts smoothly and faithfully on a manifold $M$, then one obtains a homomorphism $\Phi: G \to Diff(M)$. Then we get a map $\phi:g \to Vect(M)$, where $g$ is the Lie algebra of $G$, and $Vect(M)$ is the Lie algebra of $Diff(M)$. So one wants to classify maximal finite-dimensional Lie algebras of $Vect(M)$. First, if $g$ is not semisimple, then it has a non-trivial center $c$, which is generated by a non-zero smooth vector field $V$. One should then be able to take a quotient $M/V$ of $M$ with action on $M/V$ by the Lie algebra $g/c$ and apply an inductive argument. Actually, one should try to do this only locally, since the quotient might not be nice. Then assume $g$ is semisimple. Its Cartan subalgebra gives $R^m$ acting on $R^n$. This gives $m$ commuting vector fields, and in particular gives $m$-dimensional coordinates at a point, so $m\leq n$. Now, one needs to appeal to the classification of semisimple Lie algebras to finish off the proof (the Cartan subalgebra together with Weyl group determines the Lie algebra), and then apply the inductive argument to deal with the radical of the Lie algebra. I haven't worked out how to do this, but it seems like a plausible approach. I suspect an argument like this may be well-known in the right circles.
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.
William Goldman's article "What is a projective structure?" in the Notices of the AMS states that Daryl Cooper has shown that $\mathbb{RP}^3\#\mathbb{RP}^3$ has no $\mathbb{RP}^3$ structure. The article gives further references, but I couldn't find a reference for the Cooper result.