Understanding renormalizability and bare mass
As I said in the comment, as long as you don't hit singularities for $k^2=0$, the function $I(k^2)$ is completely regular and you can perform the expansion. As well, you can take the one-dimensional integral (it may be a simplification of a more general case you can find in computing n-point Green functions)
$$ I(k) = \int_0^{+\infty} d q \frac{q^4}{(q+k)^2} $$
where $k$ is the external momenta. This has degree of divergence $D=2$ but you can derive three-times with respect to the external momenta $k$ and you get I'''(k) which is completely convergent
$$ I'''(k) =\int_0^{+\infty} d q -6\frac{6q^2}{(k+q)^4} = -\frac{2}{k} $$
Then, you can integrate back (integrate wrt to external momenta!!)
$$ I''(k) = -2\log(k) + A $$
where A (and in the following all the upper case letters) is a divergent constant. Then,
$$ I'(k) = +2k -2k \log(k) + A k + B\rightarrow I(k) = Bk + \frac{1}{2}\left( 3+A\right)k^2 - k^2\log(k) $$
Regarding your second question, you can express the n-point Green $G^{(n)}(p)$ function in terms of the amputated one $G^{(n)}_{amp}$
$$ G^{(n)}(p_1,...,p_n) = \Pi_{i=1}^n \left[G^{(2)}(p_i)\right] G^{(n)}_{amp}(p_1,...,p_n) $$
The S-matrix is nothing else the amputated Green function in which you add the wave-function polarization and then put everything on shell. In the case of the scalar theory the wave-function polarization is trivial (i.e. it is 1). For $n=2$ you look at the full propagator $G^{(2)}(p)$ $$ G^{(2)}(p) =G^{(2)}(p)G^{(2)}(p) G^{(2)}_{amp}(p) $$
and you see that
$$ G^{(2)}(p) = \frac{1}{G^{(2)}_{amp}(p)} \,,\qquad \qquad (1) $$
(EDITED) In perturbation theory, you can set $G^{(2)}_{amp}(p) = k^2 - m^2 + a +b+k^2$. I guess the term $a +b+k^2$ include some power of the perturbative coupling constant. You can avoid higher order corrections because they are already taken into account in Eq. (1).
Notice that $G^{(2)}_{amp}(p)$ has that value because you are using the Feynman rules of the kinetic term seen as a vertex and the result of the loop-integral. If you want, the loop-integral provide you a contribution into the effective action proportional (roughly) to $(a+b)\phi^2 - (\partial\phi)^2$. Then, the term $a+b+k^2$ is the feynman rule associated to this vertex and it enters in $G^{(2)}_{amp}(p)$.
Notice I am not doing the sum like in the other answer. This is because I should do the same sum but with $1/(k^2-m^2)\rightarrow G^{(2)}(p)$ where $ G^{(2)}(p)$ is the full quantum propagator (the one which include quantum corrections). If you work in perturbation theory, that sum is equivalent to what I did here.