Is it ok to have mixed grounds and neutrals on bars in a breaker box?
The wiring issue is not a matter of pride, neatness or whatever... it's a safety issue. :)

At the service panel (ONLY AT THE SERVICE PANEL - HUGELY IMPORTANT) the neutral bus bar is bonded to ground.  You should see the ground lead and neutral tied to the same bus (the neutral bus bar).  Based on your description, it sounds like your panels are wired correctly. It just doesn't "look right" based on how the other panel is made up.  I can completely understand how this can seem incorrect from a common sense perspective. 

However, any sub-panel after the primary service from there MUST have an isolated neutral.  DO NOT DO NOT DO NOT bond neutral to ground in a sub-panel.

Why is this?  When you tie neutral to earth ground in a subpanel, you create a potential parallel path for current to return via earth (ground). In the event of a fault, your ground conductor has assumed the role of the return path for current and now everything that you've grounded (sub-panel, appliances, metal fixtures, etc) to that sub-panel is now hot.

All it takes is a preexisting fault, one rainstorm, or wet feet, whatever... and you touching something energized - and you're doing the 60 cycle shuffle.