Some questions about scalar curvature
The Kazdan-Warner theorem goes a long way toward answering the first and second questions.
(For notes typed up by Kazdan, see http://www.math.upenn.edu/~kazdan/japan/japan.pdf.)
Here's what is says (taken almost verbatim from the notes, page 93): Divide the class of all closed manifolds (edit: of dimension > 2. See comments) into 3 types:
I. Those which admit a metric of nonnegative scalar curvature which is positive somewhere.
II. Those which don't but admit a metric of 0 scalar curvature.
III. All other closed manifolds.
The theorem is that if $M$ is in class I, then any $f:M\rightarrow\mathbb{R}$ is the scalar curvature of some metric.
If $M$ is in class II, then $f:M\rightarrow\mathbb{R}$ is the scalar curvature of some metric iff it's identically 0 or negative somewhere.
If M is in class III, then $f:M\rightarrow\mathbb{R}$ is the scalar curvature of some metric iff it's negative somewhere.
In particular, every closed manifold has a metric of constant negative scalar curvature. Those in class I or II have a metric of 0 scalar curvature, and those in class I have a metric of constant positive scalar curvature.
Dear Paul Siegel,
Concerning your question 1, it is related to Yamabe problem (as it was pointed out by W. Wong). More precisely, this problems asks whether one can find a conformal deformation $g=u.g_0$ (where $u>0$ is a smooth function) of a given metric $g_0$ on a compact boundaryless manifold of dimension $n\geq 3$ such that the scalar curvature of $g$ is constant. In some sense, Yamabe problem is a higher-dimensional counterpart of Poincare-Koebe uniformization theorem (for Riemann surfaces).
Roughly speaking, the history of Yamabe problem is:
-Yamabe claimed in 1960 that he solved this problem (as a preliminary step towards Poincare conjecture), but, as it was pointed out by Trudinger, Yamabe's solution had a gap: basically, he converted the constraint "scalar curvature of $g$ is constant" into an non-linear elliptic PDE involving the conformal factor $u$, and Yamabe's assertion was that this PDE was solvable by "usual" elliptic theory, but this is wrong because the corresponding PDE has a so-called "critical" non-linearity (in the sense that this PDE is exactly a borderline case of the "standard" theory)
-The case of the round sphere was dealt with by Obata, who characterized all solutions to Yamabe PDE.
-In view of the stereographic projection, the case of the round sphere is called globally flat case.
-Aubin showed that the case of manifolds of dimension $n\geq 6$ which are not locally flat (i.e., there is some point having some neighborhood which is not conformal to the flat Euclidean space), Yamabe PDE is solvable. To do so, he construct some local test functions to prove that a quantity (related to Yamabe's PDE) called Yamabe quotient is strictly smaller than the same quotient for the round sphere. Once this is proven, the standard theory allows to conclude the solvability of Yamabe's problem.
-In the remaining cases (i.e., a compact manifold of low dimension $n=3,4,5$ or a locally (but not globally) flat manifold), Schoen replaced the use of local test functions by the use of global test function (obtained by suitable gluing of local test functions with appropriate Green functions of Yamabe PDE). Again, using these global test function, Schoen's goal was to show that the Yamabe quotient of our manifold was strictly smaller than the corresponding Yamabe quotient of the round sphere. To do so, Schoen applies a brilliant idea of expanding the Yamabe quotient in terms of the Yamabe quotient of the local function (which turns out to be almost the same of the round sphere) and then checking that the contribution of the Green function is robustly negative because of the so-called Positive Mass Theorem (of Schoen and Yau) (whose connections with General Relativity are well-known).
-After these answers to Yamabe's problem, Schoen asked about the compactness of the set of solutions of Yamabe PDE (in the non-globally flat case, of course). It turns out that this problem was completely solved by the works of Khuri, Marques, Schoen, Brendle and Brendle, Marques: assuming the Positive Mass Theorem, Schoen's question (or perhaps conjecture) has a positive answer in dimensions $3\leq n\leq 24$ but there are non-globally flat counterexamples in dimensions $n\geq 25$ (and, furthermore, this strange "dimensional" behavior is explicitly explained by the lack of definiteness of a certain bilinear forms.
For a nice introduction to Yamabe problem, please see this excellent classical paper of Lee and Parker.
A big classification result that I'm aware of is due to Gromov and Lawson.
Theorem. Let $M$ be a compact simply connected manifold of dimension $\geq 5$, which is not a spin. Then $M$ admits a metric of positive scalar curvature. A spin manifold of dimension $\geq 5$ carries a metric of positive scalar curvature iff its $\alpha$-genus is zero.
The Seiberg-Witten invariants provide special obstructions to existence of a metric of positive scalar curvature in dimension 4.
There are two good survey articles on the subject by J. Rosenberg (link 1) and S. Stolz (link 2).