Regarding question 1: In general (without additional assumptions), this statistic does not follow a t-distribution because, as you said, neither is the mean normally distributed at arbitrary sample sizes nor does the variance follow a chi-squared distribution when normality of the sample values may not be assumed (well, put aside some asymptotic considerations that do not matter for the answer of your question). The statistic is called t mainly because it is computed using a very similar formula but it does not follow a t-distribution and the text does not imply that it should. However, if normality holds, this distribution will closely resemble a t-distribution on the long run.
Regarding your question 2: Bootstrapping essentially means resampling from your sample to get an estimate of the true unknown sampling distribution of your t-statistic. Here, you estimate the sampling distribution of your t-statistic. It will be an approximate t-distribution under conditions under which the "classic" t-test would be valid and under any other conditions, it will resemble whatever the sampling distribution is. For instance, if your variable follows a skewed distribution, the bootstrap samples of t will also be skewed. Then, the t distribution is no longer your best approximate for the sampling distribution but rather your estimated distribution is. Hence, you should compute the quantiles on this distribution instead.
Does this help?