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: 
Since you presenting the hexmap through an isometric view that shifts things around.

Here is the traditional version with horizontal hexes that are effectively squished in the vertical axis to make a pseudo-isometric look.

![hex view 2](https://i.stack.imgur.com/V3CdB.jpg)

This takes a different approach with the hexes that are rotated to present no vertical or horizontal lines.

![hex view 1](https://i.stack.imgur.com/gnfGr.jpg)

Either style can work, just depends on what you want to do, the primary direction players will move through the world, and how much you want to avoid zig-zagging.
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.