The groups are ((16 high improvement+16 low improvement)+30 control), so why is it (15), t(15), t(30), and then later t(16)? Does t(n) not mean that it's a t statistic over a population of n?

Not usually. Numbers in brackets after a well-known statistic normally represent parameters for that statistic's distribution; in the case of a t-test the bracketed number would be the number of degrees of freedom, which might be one less than the sample size (for a one-sample t-test) or two less than the sum of sample sizes (for an equal variances two-sample t-test).

(Disclaimer: I haven't read the paper.)

[Edited for unambiguity.]