The dumbed down version of the central limit theorem says that regardless of the distribution of the original data, the sampling distribution of the sample mean is approximately normal, provided the sample size is fairly large. 375 is definitely large enough for your problem.
You can reasonably calculate a confidence interval for the mean using a z table here. -.11 +- 1.96*.9/sqrt(375).
( -.2011, -.0189)


"Also, how large of a sample size do I need to be "95% confident" that the true "expecation" is within +/- 0.02 of my estimated expectation. (In other words, how much data do I need to be 95% confident that the expecation is between -0.13 and -0.09?"

The first statement in here is a reasonable question, the second part is not, but I know what you mean.

The sample size required to get an estimate within .02 is going to be ballpark 7780. ( Using your estimate of the sd as the real deal ).
Also, keep in mind this is all assuming a random sample. Not "man, I've been hot as hell lately, let's do a 95
% confidence interval for my expectation".