Can I safely refreeze stock that I have brought to the boiling point?
If so, should I boil it or bring it above a certain temperature for a certain length of time to ensure food safety?
Is the presence of fats or oils a point of concern for rancidity or food safety?
Finally, are there some kinds of stock (vegetable/fish/meat/poultry) that need to be handled differently? Or will boiling suffice for any liquid preparation (soups, reductions, gastrique)?
If the stock has safely been held (under refrigeration for less than 3 days, less than 2 hours cumulative in the "danger zone" [40F - 140F, 4.5C - 60C]), then it would be considered safe before freezing or refreezing. It is always safe to freeze food that was safe to begin with.
Freezing, and especially refreezing, can sometime negatively affect quality, but I wouldn't expect a noticeable effect on broth. Freeze away, as soon as possible after making it or getting it home.
Some people freeze broth in ice trays, then store the cubes in the freezer. That way, you only pull out of the freezer what you need that day. Food that was safe when frozen and stays frozen (at 0F, -17C, normal home freezer temperature or lower), will stay safe indefinitely.
You can also speed-chill the stock by putting it in a pan in an ice-water bath, or set it on one or more of those blue-ice bricks. This will reduce the overall time in the danger zone. Reducing the broth further speeds the process by increasing the ratio of ice mass to broth.
