Homotopy groups of Diff(X) and Homeo(X)
No, this is not true, not even for spheres. Consider the following commutative diagram: $\require{AMScd}$ \begin{CD} \text{Diff}_{\partial}(D^d) @>>> \text{Homeo}_{\partial}(D^d) \sim *\\ @V f V V @VV V\\ \text{Diff}(S^d) @>>g> \text{Homeo}(S^d) \end{CD} We have that $\text{Homeo}_{\partial}(D^d)$ is contractible by the famous Alexander trick. Now if $d \geq 5$ is odd, it is known that $\pi_{\ast} \text{Diff}_{\partial}(D^d) \otimes \mathbb Q$ is often non-trivial. In degrees $4i-1$ below some range growing with the dimension this was discovered by Farrell and Hsiang; a recent improvement was found by Krannich. But there are also other classes, see Watanbe: On Kontsevich's characteristic classes for higher‐dimensional sphere bundles II: Higher classes, for instance.
In any case, there are many non-trivial classes in rational homotopy that die under $g \circ f$. As $f$ can be seen as the fiber of $t\colon \text{Diff}(S^d) \to Fr(S^d) \sim O(d+1)$ where $Fr(S^d)$ is the frame bundle of $S^d$ and $t$ sends a diffeomorphism to the differential of the north pole of $S^d$ and there is the obvious section coming from the action of $O(d+1)$ and $S^d$, we see that these classes all survive under $f_{\ast}$. Hence they lie in the kernel of $g_{\ast}$.
No, the statement about the kernel and cokernel being finite is not true.
For a closed $d$-manifold, $d \neq 4$, smoothing theory identifies the homotopy fibre of $$B\mathrm{Diff}(M) \longrightarrow B\mathrm{Homeo}(M)$$ with (certain path components of) the space of sections of a bundle $$Top(d)/O(d) \longrightarrow E \longrightarrow M$$ constructed from te tangent bundle of $M$. Supposing for simplicity that $M$ is parallelisable, this space of sections is equivalent to $$\mathrm{map}(M, Top(d)/O(d)).$$
So your question is more or less equivalent to asking about the homotopy groups of $Top(d)/O(d)$. Until recently it was not even known whether the homotopy groups of $Top(d)/O(d)$ are finitely-generated, but that was proved by Kupers, in Some finiteness results for groups of automorphisms of manifolds. These days quite a bit is known about the rational homotopy groups of these spaces. Using the above formulation of smoothing theory (which goes through unchanged for manifolds with boundary) for the disc $D^d$, you will find most results are stated for $$B\mathrm{Diff}_\partial(D^d) \simeq \Omega^d_0(Top(d)/O(d)).$$ (Here I have used that $B\mathrm{Homeo}_\partial(D^d) \simeq *$ by the Alexander trick.) Any result which shows that $B\mathrm{Diff}_\partial(D^d)$ has a rationally nontrivial homotopy group provides a counterexample to your finiteness question.
For example:
$\pi_i(Top(2n)/O(2n)) \otimes \mathbb{Q}=0$ for $i< 4n-2$, but $$\pi_{4n-2}(Top(2n)/O(2n)) \otimes \mathbb{Q}= \mathbb{Q}.$$ (See Kupers--Randal-Williams On diffeomorphisms of even-dimensional discs.)
$\pi_i(Top(2n+1)/O(2n+1)) \otimes \mathbb{Q}=0$ for $i< 2n+5$, but $$\pi_{2n+5}(Top(2n+1)/O(2n+1)) \otimes \mathbb{Q}= \mathbb{Q}.$$ (This follows from Farrell--Hsiang On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds.)
There are some slides from a talk I recently gave at https://www.dpmms.cam.ac.uk/~or257/MIT2020.pdf, which give some more detail about the state of the art.