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.
After thinking about this question for two hours, my belief is strengthened that there cannot be a proof of the vanishing of the A-genus
for spin manifolds with positive scalar curvature without using both the index theorem for the Dirac operator and the Lichnerowicz formula (or other analytic techniques).
In the comments to the question, a proof using the integrality of the $\hat{A}$-genus and Chern-Weil theory has been suggested. I wish to argue
that there is absolute no reason to believe such a proof might work out.
Objections to the use of the integrality theorem.
The manifold $CP^2$ has positive scalar curvature and $\hat{A}$-genus $1 /16$. It is of course not spin. Take the connected sum
of 16 copies of $CP^2$ and call it $M$. The $\hat{A}$-genus of $M$ is $1$; $M$ is still not spin. The famous surgery lemma by Gromov and Lawson
implies that $M$ has a metric of positive scalar curvature, because $M$ is obtained from the manifold $\coprod_{16} CP^2$
by a sequence of surgeries of codimension $4 \geq 3$. So there are connected pscm manifolds with integral but nonzero $\hat{A}$-genus. More generally,
it can be shown that any simply connected, high-dimensional, oriented and non-spin manifold admits a metric of positive scalar curvature. This also
goes back to Gromov and Lawson,
and Sven F\"uhring worked out the details in his thesis:
http://www-m10.ma.tum.de/bin/view/Lehrstuhl/SvenFuehring#EigenePublikationen
This result shows that the integrality of the $\hat{A}$-genus has nothing to do with positive scalar curvature and that the spin hypothesis has to be used in a more subtle way.
Objections to the use of Chern-Weil theory:
At a first glance, it seems tempting to use Chern-Weil theory for such questions, because it expresses characteristic classes in terms of curvature. Moreover,
a conceptual proof of the Gauss-Bonnet theorem can be given with help of this theory. But an examination of the argument quickly shows that this
is a lucky accident. The first accident is that the Euler class has a direct
geometric-topological meaning, which allows to relate it to another topological invariant (the Euler number), while the complicated rational polynomial in the Pontrjagin classes has no such meaning
(Paul Siegel alluded to this point in his comment).
The second accident is that the integrand of the Gauss-Bonnet-Chern theorem only depends on the curvature tensor and not on the metric itself
(though to define the integrand,
one needs to know that the connection preserves some metric).
The problem with scalar curvature is that it depends on both, the connection and the metric, while the Chern-Weil forms only depend on the connection and not on the metric.
Of course there is a universal formula for the $\hat{A}$-form of any Riemann manifold $(M,g)$ of the form
$\hat{A} = f(jet^2 g) \cdot vol_g$, where $f$ is an algebraic function of the $2$-jet of the metric. I have not worked out the concrete formula for $f$;
for $4$-dimensional manifolds, this should not be too complicated and might be a worthy exercise. I predict that you won't find a formula for $f$
that depends on the scalar curvature in a controllable way. Another remark is that if such a formula gives a result for positive scalar curvature,
it should give a result for negative scalar curvature as well. On the other hand, any manifold of large dimension has a metric of negative scalar curvature (Lohkamp).
The conclusion I draw is that Chern-Weil theory is insensitive to the sign of scalar curvature. The next problem is that Chern-Weil theory is
insensitive to the spin condition as well: the map $Sym^{\ast} (\mathfrak{so}_n)^{SO(n)}\to Sym^{\ast} (\mathfrak{spin}_n)^{Spin(n)}$
on the space of invariant polynomials is an isomorphism. In other words, the spin condition does not yield any relation between Chern-Weil classes. The integrality of the $\hat{A}$-genus cannot be obtained
by a local consideration; this would equate it to another manifestly integral class, which is not the case.
The view that positivity should enter the proof in a more global way is furthermore supported by Gromov's h-principle, a special case of which states that
any open manifold (no compact component) has a metric of positive scalar curvature. This implies
that any obstruction to positive scalar curvature has to arise in an essential global way.
My conclusion of the above discussion is that the integrality result forgets the relevant information from the spin condition and that
Chern-Weil theory is not the
correct recipient for the information coming from the positivity of the curvature. Philosophically, integrality is too global to capture the spin condition and Chern-Weil theory is too local to capture the sign of the curvature.
It is a pity that Misha Gromov is not active on this site.....
EDIT: here is an example which shows that any hope to use Chern-Weil theory to obtain information on scalar curvature is in vain.
Fact 1: Let $\nabla_g$ be the Levi-civita connection of the Riemann manifold $(M,g)$, then for all $c>0$, $\nabla_{cg} = \nabla_g$.
Fact 2: the scalar curvature of $(M,cg)$ is $c^{-1} scal_g$. Fact 3: the scalar curvature of a product is the sum of the scalar curvature.
Now take $S^2$, $g_0$ the standard metric with $scal_{g_0}=1$; $F$ a genus $\geq 2$ surface with metric $g_1$ and $scal_{g_1}=-1$.
Then the product $M=S^2 \times F$ with product metric $(c_0 g_0 )\times (c_1 g_1)$ has scalar curvature $c_{0}^{-1} - c_{1}^{-1}$. Suitable choices of $c_i$ yield any real value for the scalar curvature. But by fact 1, the Levi-Civitta connection and hence the Chern-Weil invariants do not depend on $c_i$.
Best Answer
The area in dimension $2$ for the sphere (with abitrary metric): In C. Bär: Lower eigenvalue estimates for Dirac operators (Math. Ann. 293) it was shown that $$\lambda^2 area\geq 4\pi.$$ This can be generalized to higher genus surfaces if one restricts to eigenvalues with higher multiplicity.