Personal comment: It seems the discussion in this question finally led me to understand how to modify Agol's argument to answer the present question. In fact, my motivation when asking that question answered by Agol was mostly to resolve the present question.
Answer: For $M$ a non-compact manifold without boundary, it seems the homotopy fibre of the inclusion $\iota:\textrm{Diff}(M)\hookrightarrow\textrm{Emb}(M,M)$ at $\textrm{id}_M$ is weakly contractible. As hinted implicitly by Tom Goodwillie in his answer, it is not hard to conclude that every homotopy fibre of $\iota$ is then either empty or weakly contractible.
Before continuing, I will clarify the structure of the homotopy fibre, $F$, of $\iota:\textrm{Diff}(M)\to\textrm{Emb}(M,M)$ at $\textrm{id}_M$. Recall from the question above that $\textrm{Diff}(M)$ and $\textrm{Emb}(M,M)$ are both endowed with the compact-open (or weak) $C^1$-topology. As a set, $F$ consists of the continuous isotopies $(\varphi_t:M\to M)_{t\in I}$ through embeddings $M\to M$ for which $\varphi_0 = \textrm{id}_M$, and $\varphi_1$ is a diffeomorphism of $M$. An isotopy $(\varphi_t)_{t\in I}$ will also be written as a map $\varphi : M\times I \to M$. As a topological space, $F$ is the subspace of the space of paths in $\textrm{Emb}(M,M)$ consisting of the paths which start at $\textrm{id}_M$ and end at a diffeomorphism of $M$.
Further, let me state a strong form of the isotopy extension lemma which I will require later: when $X$ is a manifold without boundary, and $Y$ is a compact manifold, the map
$$ (\textrm{eval},\textrm{proj}):\textrm{Diff}_c(X)\times\textrm{Emb}(Y,X) \longrightarrow \textrm{Emb}(Y,X)\times\textrm{Emb}(Y,X) $$
given by $(\textrm{eval},\textrm{proj})(u,v)=(u\circ v,v)$ is a fibration. Here $\textrm{Diff}_c(X)$ denotes the space of compactly supported diffeomorphisms of $X$, equipped with the strong (or Whitney) $C^1$ topology --- the use of the strong topology is essential in the present answer, as remarked at a later point. The map $(\textrm{eval},\textrm{proj})$ is a locally trivial fibre bundle, as one can give local trivializations using tubular neighbourhoods; thus the above map is also a fibration, since the base is paracompact (even a metrizable space). The above statement implies most usual forms of the isotopy extension lemma.
I will now describe the argument that the homotopy fibre $F$ of $\iota:\textrm{Diff}(M)\to\textrm{Emb}(M,M)$ at $\textrm{id}_M$ is weakly contractible. It is a modification of Agol's ingenious answer linked above.
Fix a map $f:K\to F$ where $K$ is compact. I will describe a homotopy between $f$ and the constant map $b:K\to F$ equal to the basepoint $b=(\textrm{id}_M)_{t\in I}$ of $F$. First we construct an exhaustion of $M$ by compact submanifolds $M_i$ for $i\in {\mathbb N}$ such that:
- $\bigcup_{i\in {\mathbb N}} M_i = M$;
- $M_0=\emptyset$;
- $M_i \subset \textrm{int}\ M_{i+1}$ for all $i\in {\mathbb N}$;
- $\varphi(M_i \times I)\subset \textrm{int}\ M_{i+1}$ for all $\varphi$ in the image of $f$.
For example, we may use a proper smooth function $\rho: M\to [0,+\infty)$, and take $M_i=\rho^{-1}([0,x_i])$ for a suitable unbounded, strictly increasing sequence $(x_i)_{i\in{\mathbb N}}$ of regular values of $\rho$ (which are dense in $[0,+\infty)$ by Sard's theorem). The final condition above is where the compactness of $K$ is required, and I do not know how to avoid using compactness there.
With the compact submanifolds $M_i \subset M$ at hand, we can inductively construct a homotopy $f\simeq b$. Set $f_0=f$, and let $n\in {\mathbb N}$. Assume inductively that we have constructed, for each $k < n$, a map $f_{k+1}:K\to F$, and a homotopy $H_k : f_k \simeq f_{k+1}$. Furthermore, assume that for all $\phi$ in the image of $f_{k+1}:K\to F$, and for all $\psi$ in the image of $H_k: K\times I\to F$, the following conditions hold:
- $\phi_t|_{M_k} = \textrm{id}_{M_k}$ for all $t\in I$;
- $\psi_t|_{M_{k-1}} = \textrm{id}_{M_{k-1}}$ for all $t\in I$;
- $\psi(M_l \times I) \subset \textrm{int}\ M_{l+1}$ for each $l\in {\mathbb N}$ with $l\geq k$.
Now we construct a homotopy $H_n : f_n \simeq f_{n+1}$ based on the above data. Let $\varphi = f_n(x)$ for a fixed $x\in K$. By condition (3) (and $M_0=\emptyset$), we have $\varphi(M_n \times I) \subset \textrm{int}\ M_{n+1}$. Therefore, the restriction $\varphi|_{M_n \times I}$ is an isotopy through embeddings $M_n \to \textrm{int}\ M_{n+1}$. Since $\varphi_0 = \textrm{id}_M$, the isotopy extension lemma implies that $\varphi|_{M_n \times I}$ extends to a compactly supported diffeotopy $\widetilde{\varphi}=g(x): (\textrm{int}\ M_{n+1})\times I \to \textrm{int}\ M_{n+1}$ with $\widetilde{\varphi}_0=\textrm{id}_{\textrm{int}\ M_{n+1}}$.
To guarantee that $\widetilde{\varphi}=g(x)$ depends continuously on $x\in K$, we now take a detour in which we apply the strong form of the isotopy extension lemma stated earlier. Consider the commutative square
$$ \begin{matrix}
K & \xrightarrow{\ (a,b)\ } & \textrm{Diff}_c(\textrm{int}\ M_{n+1})\times\textrm{Emb}(M_n,\textrm{int}\ M_{n+1}) \\
\llap{\scriptstyle \textrm{incl}_0}\Big\downarrow & & \llap{\scriptstyle (\textrm{eval},\textrm{proj})}\Big\downarrow \\
K\times I & \xrightarrow{\smash{\ (c,d)\ }} & \textrm{Emb}(M_n,\textrm{int}\ M_{n+1})\times\textrm{Emb}(M_n,\textrm{int}\ M_{n+1})
\end{matrix} $$
where the components of the horizontal maps are defined by
$$ a(x) = \textrm{id}_{\textrm{int}\ M_{n+1}} \qquad b(x) = [f_n(x)]_0|_{M_n} = ( \textrm{incl}:M_n\hookrightarrow \textrm{int}\ M_{n+1} ) $$
$$ c(x,t) = [f_n(x)]_t|_{M_n} \qquad d(x,t) = b(x) = ( \textrm{incl}:M_n\hookrightarrow \textrm{int}\ M_{n+1} ) $$
(so only the map $c$ is not constant). Since the vertical map on the right of the square is a fibration, there exists a diagonal lift
$$ (e,d) : K\times I \longrightarrow \textrm{Diff}_c(\textrm{int}\ M_{n+1})\times\textrm{Emb}(M_n,\textrm{int}\ M_{n+1}) $$
and the map $g:K\to F$ (such that $\widetilde{\varphi}=g(x)$) is determined by
$$ \widetilde{\varphi}_t=[g(x)]_t = e(x,t) $$
It is easy to check the required conditions:
- $\widetilde{\varphi}=g(x)$ and $\varphi=f_n(x)$ coincide on $M_n \times I$;
- $\widetilde{\varphi}_0=[g(x)]_0=\textrm{id}_{\textrm{int}\ M_{n+1}}$.
as follows from the fact that $(e,d)$ is a diagonal lift for the square above.
In conclusion, we have a compactly supported diffeotopy $\widetilde{\varphi} = g(x)$ of $\textrm{int}\ M_{n+1}$ (depending continuously on $x\in K$) which extends $\varphi|_{M_n \times I}$. Extend $\widetilde{\varphi} = g(x)$ to all of $M$ by making it the identity outside $\textrm{int}\ M_{n+1}$. Fundamentally, observe that this extension operation gives a continuous map $\textrm{Diff}_c(\textrm{int}\ M_{n+1}) \to \textrm{Diff}_c(M) \to \textrm{Diff}(M)$ thanks to the strong (or Whitney) $C^1$ topology we placed on the space of compactly supported diffeomorphisms, so we are preserving continuity on $x\in K$. Then $\widetilde{\varphi} = g(x)$ becomes a compactly supported diffeotopy of the whole manifold $M$ depending continuously on $x\in K$. Moreover, $\widetilde{\varphi}_0 = \textrm{id}_M$.
It is now easy to define $f_{n+1} : K \to F$:
$$ [f_{n+1}(x)]_t = (\widetilde{\varphi}_t)^{-1} \circ \varphi_t = ([g(x)]_t)^{-1} \circ [f_n(x)]_t $$
Condition (1) above then follows from the fact that $\widetilde{\varphi}$ extends $\varphi|_{M_n \times I}$. Note that $f_{n+1}$ is continuous since mapping a diffeomorphism to its inverse is continuous in the compact-open $C^1$ topology ($\textrm{Diff}(M)$ is a topological group); alternatively, we can instead use that $g$ is actually continuous with respect to the strong $C^1$ topology on $\textrm{Diff}_c(M)$, and that inversion is also continuous in the strong $C^1$ topology.
We further define the homotopy $H_n : K\times I \to F$ between $f_n$ and $f_{n+1}$ as follows:
$$ [H_n(x,s)]_t = (\widetilde{\varphi}_{\min\{s,t\}})^{-1} \circ \varphi_t = ([g(x)]_{\min\{s,t\}})^{-1} \circ [f_n(x)]_t $$
As was done for $f_{n+1}$, we can show that $H_n$ is continuous. It is also easy to check that $H_n$ takes values in $F$: $[H_n(x,s)]_0 = (\widetilde{\varphi}_0)^{-1} \circ \varphi_0 = \textrm{id}_M$, and $[H_n(x,s)]_1 = (\widetilde{\varphi}_s)^{-1} \circ \varphi_1$ is a diffeomorphism of $M$. Moreover, the conditions (2) and (3) above are also straightforward:
- Condition (2): Note that $\varphi_t|_{M_{n-1}} = [f_n(x)]_t|_{M_{n-1}} = \textrm{id}_{M_{n-1}}$ (which is condition (1) for $f_n$). Since $\widetilde{\varphi}$ extends $\varphi|_{M_n \times I}$, we also have that $\widetilde{\varphi}_t|_{M_{n-1}} = \textrm{id}_{M_{n-1}}$. We calculate
$$ [H_n(x,s)]_t|_{M_{n-1}} = (\widetilde{\varphi}_{\min\{s,t\}})^{-1} \circ \varphi_t|_{M_{n-1}} = (\widetilde{\varphi}_{\min\{s,t\}})^{-1}|_{M_{n-1}} = \textrm{id}_{M_{n-1}} $$
- Condition (3): Fix $l\in {\mathbb N}$ with $l\geq n$. We know that $\varphi(M_l\times I)\subset \textrm{int}\ M_{l+1}$ (either by condition (3) above if $n > 0$, or by the last condition for the submanifolds $M_i$ if $n=0$). Hence
$$ [H_n(x,s)]_t (M_{l}) = (\widetilde{\varphi}_{\min\{s,t\}})^{-1} \circ \varphi_t (M_l) \subset (\widetilde{\varphi}_{\min\{s,t\}})^{-1} (\textrm{int}\ M_{l+1}) = \textrm{int}\ M_{l+1} $$
where the last equality is a consequence of $\widetilde{\varphi}_u$ being a diffeomorphism which is the identity outside $\textrm{int}\ M_{n+1}$ (and $l\geq n$).
This completes the inductive construction of $H_n : f_n \simeq f_{n+1}$.
To finish the proof, consider the infinite concatenation of all the homotopies $H_n$:
$$ H = H_0 \ast ( H_1 \ast ( H_2 \ast ( H_3 \ast \cdots ) ) ) $$
i.e. $H$ is:
- $H_0$ at twice the speed on the interval $[0,\frac 1 2]$;
- $H_1$ at four times the speed on the interval $[\frac 1 2,\frac 3 4]$;
- $H_2$ at eight times the speed on the interval $[\frac 3 4,\frac 7 8]$;
- in general, $H_n$ at $2^{n+1}$ times the speed on the interval $[\frac{2^n-1}{2^n},\frac{2^{n+1}-1}{2^{n+1}}]$.
Finally, we declare $H(-,1)=b$. Then $H$ is the desired homotopy $H : f \simeq b$. The fact that $H$ is continuous at time $1$ is a consequence of:
- most importantly, the above condition (2) for the homotopies $H_k$;
- the submanifolds $M_i$ form a compact exhaustion of $M$ (their union is $M$, and $M_i \subset \textrm{int}\ M_i$);
- together with the fact that we are considering the compact-open $C^1$ topology on the space of embeddings $\textrm{Emb}(M,M)$.
This concludes the proof.
Best Answer
Regarding question 1 the following is well known.
For a reasonably nice space $X$ (say, a finite connected CW-complex) let $Aut_0(X)$ be the monoid of self-equivalences of $X$ homotopic to the identity and let $Aut_0^\bullet(X)$ be the submonoid fixing a basepoint. Then there is a natural fibration given by the evaluation map $Aut_0^\bullet(X)\to Aut_0(X)\to X$. This gives rise to the fibration $X\to BAut_0^\bullet(X)\to BAut_0(X)$ which is the universal fibration with fiber $X$.
For $X=S^n$ it's easy to see that $Aut_0^\bullet(S^n)\cong Map_0^\bullet(S^n, S^n)\cong Map_0^\bullet(S^{n-1}, \Omega S^n)\cong \ldots Map_0^\bullet (S^0, \Omega^n(S^n))=\Omega^n(S^n)$. In particular, $\pi_k(Aut_0^\bullet(S^n))\cong \pi_{n+k}(S^n)$. Combining this with the fibration $Aut_0^\bullet(S^n)\to Aut_0(S^n)\to S^n$ this in principle gives you homotopy groups of $Aut_0(S^n)$. In practice though I don't think this is very useful away from rational coefficients or for small values of $k$ where you have stability.
I'm not familiar with the general theory for spaces other than spheres. As I mentioned in my comment, rationally the situation is in principle well understood. But even for rational coefficients the homotopy structure of $Aut_0(X)$ is complicated when $X$ is a wedge of spheres. As I mentioned, rational homotopy groups of $Aut_0(X)$ grow exponentially. See "The monoid of self-homotopy equivalences of some homogeneous spaces" by Félix and Thomas for a careful proof of that. One general fact that I know is a result of Dror, Dwyer and Kan that when $X$ is a nilpotent finite $CW$-complex then $BAut_0(X)$ has finite type, i.e. it has the homotopy type of a CW complex with a finite number of cells in each dimension.