For 2) I think the following is an answer: Suppose $K$ is a compact Lie subgroup
of $\mathrm{Diff}(S^n)$ of dimension $\geq{n+1\choose 2}$. Being compact it is
the group of isometries of some Riemannian metric of $S^n$ and we fix one such
metric. The stabiliser of a point therefore has dimension at most $n\choose 2$
(the dimension of the orthogonal group of $\mathbb R^n$) and as an orbit has at
most dimension $n$, $K$ has at most dimension $n+{n\choose 2}={n+1\choose 2}$
and hence we have equality. This means that the stabiliser contains
$\mathrm{SO}_n$ and $K$ acts transitively. In particular the metric fixed by $K$
has constant curvature and then the curvature is positive and thus up to a
constant conformal factor is conjugate to the standard metric. This gives a
conjugation of $K$ into the isometry group of the standard sphere.

**Addendum**: Tom's example gives that the dimension of a non-compact group
acting (faithfully) on $S^n$ is unbounded. One question seems to remain namely
if the dimension of a semi-simple group action is bounded or not. By the above
we get a bound on the dimension of a maximal compact subgroup. In many cases
this seems to bound the dimension of the group itself. For instance, is the
dimension bounded if the center is finite? In that case the situation is
completely described by a Cartan decomposition of the Lie algebra and the
question is whether the $-1$-part (usually denoted $\mathfrak p$) of it has
dimension bounded by the dimension of the $+1$-part (the Lie algebra of the
maximal compact). I myself don't know enough of the real Lie group theory to
decide that.

An example beyond that is the universal cover $G$ of $\mathrm{SL}_2(\mathbb R)$.
It is contractible and has no non-trivial compact connected subgroup. It also
acts on the universal cover of $S^1$ (compatibly with the projective action of
$\mathrm{SL}_2(\mathbb R)$ on $S^1$) and hence acts continuously on $[-1,1]$
(say) fixing the endpoints. However, the action at the endpoints is not flat so
this action does not extend to a smooth action on $\mathbb R$ acting as the
identity outside if $[-1,1]$. If it could be modified to do so one could use
Tom's argument to get an action of any finite product of copies of $G$ (at least
on $S^1$). One could try to find three vector fields with support in $[-1,1]$
fulfilling the defining relations of $\mathfrak{sl}_2(\mathbb R)$ but it looks
tricky to me.

**Addendum 1**: To be more precise about the action on $[-1,1]$ we may use
$\tanh$ as diffeomorphism from $\mathbb R$ to $(-1,1)$. The rotation group of
$\mathrm{SL}_2(\mathbb R)$ lifts to the group of translations of $\mathbb R$ and
they correspond under this diffeomorphism to $\varphi_\lambda(t)=(\lambda t+1)/(t+\lambda)$ of
$(-1,1)$ for $\lambda>1$ or $\lambda<-1$ (we miss the identity map which corresponds to
$\lambda=\pm\infty$). Now, one possibility of getting an action of $G$ on $\mathbb R$ which
is the identity outside of $(-1,1)$ is by trying to conjugate the one we have by
a diffeomorphism $\gamma$ of $(-1,1)$. This requires every conjugate
diffeomorphism to be *flat* at $\pm1$ where a diffeomorphism $\psi$ of
$(-1,1)$ is flat at $1$ if $\psi(x)=x+\mathcal{O}(|x-1|^n)$ for all $n$ and $x$
close to $1$ (and similarly for $-1$). Now suppose that we have a $\gamma$ such that
conjugating the given action of $G$ by it gives an action all of whose elements
are flat. In particular we have $\gamma(\varphi_\lambda(\gamma^{-1}(x)))=x+\mathcal{O}(|x-1|^n)$
for all $n$. In particular, putting $n=2$ we get that
$\lim_{t\to1}\gamma'(\varphi_\lambda(\gamma^{-1}(t)))\varphi_\lambda'(\gamma^{-1}(t))\gamma'(\gamma^{-1}(t))^{-1}=1$
and putting $s=\gamma^{-1}(t)$, using that $\lim_{t\to1}\gamma^{-1}(t)=1$ and that
$\lim_{s\to1}\varphi_\lambda'(s)=\varphi_\lambda'(1)=(\lambda-1)/(\lambda+1)$ this gives
$\lim_{s\to1}\gamma'(\varphi_\lambda(s))/\gamma'(s)=(\lambda+1)/(\lambda-1)$. This puts a very stringent
condition on $\gamma$ and then also the same condition must be fulfilled for
$\varphi_\lambda$ replaced by any element of $G$ (as well as needing the flatness
condition for all orders $n$).