Question: What angles and long-side/short-side ratios give the most aesthetically pleasing and graphically regular isometric (squashed and flat side up) hexes, that additionally resolve to whole pixel sizes for several sizes when rendered?

Answer: 
By "flat side up", I assume you mean "flat side horizontal" (as "up" could mean either the edge itself points up or it "faces" up, i.e. its normal points up).

I experimented with both orientations for a game I developed in college. Personally, I found ["flat side vertical"](http://startreksupremacy.com/blog/galactic_supremacy.jpg) hexagons more pleasing on the eyes. I could certainly be in the minority here, but I believe Civilization V uses the same orientation, so I'm definitely not alone. If you're not dead set on using one orientation over the other, then I suggest you experiment with both. Since most of the [online resources](http://www-cs-students.stanford.edu/~amitp/gameprog.html#hex) I've found regarding hex grids use the "flat side horizontal" orientation, many equations you come across may require adjustments; [this site](http://playtechs.blogspot.com/2007/04/hex-grids.html) should help.

I built my project with a resolution-independent UI framework that performed device pixel snapping automatically, so I didn't spend much time tweaking the angles and ratios. I believe each of my hexes had a bounding box of 96 device independent pixels squared (96 device pixels on a standard 96dpi display with scale = 1.0). You should be able to derive the rest from the screenshot :).
These are for flat side horizontal. The terminology I am using is from Amit's thoughts on grids page, with the additional language of "narrow side length" meaning the length of the squished non-horizontal sides.

<http://www-cs-students.stanford.edu/~amitp/game-programming/grids/hex-grid-metrics-labeled.png>

equilateral (0 degree projection):  

"height" = L√3  

"wide width" = L  

"narrow width" = L√3  

narrow side length = L  

45 degree projection:  

"height" = L  

"wide width" = L  

"narrow width" = 2 \* √(3/8)L  

narrow side length = √(2)L  

60 degree projection:  

"height" = (3/2)L  

"wide width" = L  

"narrow width" = 2 \* √(3/4)L  

narrow side length = √(13/16)L  

I calculated these by hand, so please check my work.