The first five frets are sharp by 10-15 cents, and more on the higher three strings, so it makes me think it is the string height causing them to stretch too much when pushed down. I checked the fret positions and they seem legit.
The only amp I have is the cheap one that comes in a squier kit, but it has an earth. I've also gone straight into the line in on the computer. The squier doesn't buzz on either of those. The humbucker buzzes more so it seems like an earth problem rather than picking up noise.