My framing table does not list 2x4 for low slope (<3/12) rafters. It looks like 2x6 white pine is good for an 8' span, 2x8 for about 11'. That's 24" OC. You could probably get by with 2x6 Douglas fir or yellow pine IF you go 16" OC.
If it was my shed, I'd go with 2x8s, 24" OC. 2x4 purlins every 4'. And I'd go with a pair of 2x8s for headers (aka girders) between the posts to support the rafters. Secure the rafters to the headers with "hurricane" clips.
Your posts should be treated for ground contact. The 4x4s sold at the big box stores are ALMOST NEVER treated for ground contact. Go to a lumberyard. I recommend you go with 4x6 posts rather than 4x4; they're generally of better quality.
In most parts of the country, 42 inches is deep enough to get below the frost line. Dig your holes at least 8 inches deeper than you need, and pour a concrete plug in the bottom to support the posts.
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