Why correlation length diverges at critical point?

It is not the correlation length of the system that you should look at, but the correlation of the fluctuations. If T>>Tc the spins are randomly oriented and the lenghtscale of fluctuations is very small. As you get closer to Tc, the fluctuations become more correlated, and lenghtscale increases toward infinity. Similarly for the ferromagnet at temperatures much less than Tc, all spins are aligned. The fluctuations at 0 < T << Tc have short correlation lengths. As you heat the system, it is still mostly ordered, but the number of spins pointing in the opposite direction increases, and so does the correlation length of these fluctuations

I think your trouble is that a correlation length $\xi$ is not to be interpreted as correlation in the sense of statistics, e.g.

$ \frac{<(s(x)-\lt s(x) \gt)(s(y)-<s(y)>)>}{\sqrt{<\left( s(x)-\lt s(x)>\right)^2 \lt(s(y)-\lt s(y)>)^2 \gt)}}$,

but rather defined via $ <(s(x)-< s(x)>)(s(y)-<s(y)>)>=e^{-|x-y|/\xi}$

( see for example (https://physics.stackexchange.com/q/59690).

Assume that, at zero temperature, all spins are "frozen" and perfectly aligned and hence perfectly correlated (in the statistical sense). However, since $s(x)=<s(x)>$ and $s(y)=<s(y)>$ in this case, it follows that $\xi=0$ .

As far as the second part of the question is concerned: Critical points are phase transitions that correspond to fixed points in the renormalization group flow. What this means is that the process of consecutively dividing the spin lattice into blocks, integrating them out and constructing a new Hamiltonian between those blocks has reached a fixed point: The form of the Hamiltonian does not change any longer, only its parameters (couplings) get re-adjusted with any further block-spin operation. This in turn means that the system has lost its scale and has become scale free. So if I were to take two pictures of the material, one of size one inch and the other one of size one micro-inch, you could not tell me which one is which. The only way to describe this mathematically is by assuming a power-law which yields $\xi(T) \sim (T-T_c)^{-\nu}$ where $T_c$ is the critical temperature and $\nu$ is the scaling dimension that is not necessarily integer.Hence the correlation length diverges at the critical point.