Does it make sense to describe the SEM as an estimate? If one calculates a standard error of the mean from the sample mean and sample standard deviation of n observations drawn from a normally distributed population, does it make sense to call that SEM an estimate of some "true" SEM?
Thus, if I increase n, is it true that I not only reduce the size of my SEM, but also increase the accuracy of my estimate (because the sample standard deviation is better estimate)?

The standard error of the mean is an estimate. Specifically, it is an estimate of the standard deviation of the sample mean, seen as a random variable.
And of course, as you increase your sample size $n$, your sample mean will become less and less variable - which is exactly reflected in the SEM getting smaller.
