You are right on both counts.
The time travelling will be left for your GR courses. Yes, that was a typo, now fixed. Thank you for catching it!
Yes, the peak in the density of states is not just a peak, but rather a real divergence (which indeed is a very strong peak!). The argument given by the tutor is also correct -- the divergence is logarithmic, so it is somewhat weak, and it is of course integrable (the integral of the density of states remains finite, as it should since the total number of states in a band is fixed).
In a bit more detail, given the energy E(kx,ky) as a function of kx,ky, one can always write the density of states as
DOS(epsilon) = (L/2pi)^2 integral dkx dky delta(epsilon - E(kx,ky))
with delta being a delta function. Each state in the zone gives a contribution to the density of states at the appropriate energy. You can then try to explicitly do this integral. Do one integral (say ky) explicitly and then you will see that the remaining integrand is peaked for kx and E both close to zero. Expand for small E and small kx and then you can see the log divergence explicitly.