What exactly counts as an 'unofficial transcript' for graduate school admissions? At my school, I can request an official transcript to be emailed to me, and it comes addressed to me at the top, with a COPY watermark all over it. Since this doesn't go directly to the graduate school, would this count as an unofficial transcript? It contains the classes taken, my grades and so on.

One thing though, is that at my institution, first year grades don't 'count' per se, so they don't appear on the official transcript. Instead, these are all listed as SC [Satisfactorily Completed]. Would seeing this on the official transcript be odd to a grad school admissions committee?
An "unofficial" transcript is generally considered anything that doesn't bear the "sign and seal" (or equivalent) of the issuing institution, as well as any reproduction of the official transcript (scan, photo, copy, etc.). So your "copy" would be sufficient as an unofficial transcript.

The question about "satisfactory completion" of first-year courses is not really a big deal. Different schools know that different universities have their own systems in deciding what does and does not get graded (some schools are pass/fail for a semester, others for a full year, others allow different courses across the four years to be graded that way). So long as you're not doing that with your major courses, there's probably little harm. Moreover, freshman year is the year where little stigma will be attached if letter grades aren't reported.