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Just realized I mistyped the starting month (by one month) of a job I held 8 years ago on my resume (I'm 30 so it was my first "real" job). It's correct on the employment section of the app. Should I email a corrected version or let it slide?

Generally small typos aren't worth the correction, esp. with the PS/DS. But a resume typo can be seen as "inaccurate information". I would have e-mail them and explain that one of your entries is off by one month.