So would it make sense to think of a microstate as a region of phase space, a point and those points "very close to" it? And "increasing number of microstates" just means a larger number of these regions have non-negligible probabilities? In continuous terms you would see this as the distribution flattening out. I might be having trouble visualising what we're integrating, since if it's a probability the integral over the whole phase space can only be 1, right?
Yes, that is the principle. The probability of a single point is zero because an integral over a point is zero, hence “very close to it” (in an infinitesimal volume around the point).
The integral of the probability over the phase space is indeed 1. This is the purpose of the partition function, which is the normalisation factor. The weight function is not normalised a priori.
