Should I let stock cool with the bones/veg still in it?
You need to strain the stock and cool it until it's 40 degrees F. or below before you refrigerate it.

Leaving it to cool overnight on the stove is going to create a bacteria cesspool.  The temperature danger zone is between 40 and 140 degrees.  This is the range that is prime for the growth of bacteria.  Food safety guidelines require that it cool to 70 degrees or below within 2 hours and then below 40 degrees within another 4 hours time.  Total maximimum time that it should take to get to 40 degrees is 6 hours.

Even if you get it cooled down now with all of the vegetbles and bones in it, the problem will be that you have to heat it back up in the morning to strain it as it's going to gelatinze overnight (provided it's been cooked long enough to extract a good amount of gelatin - usually about 4-6 hours for a good chicken stock and 12-16 hours for veal/beef stock).  When it's reheated you're running the risk that the carrots and other vegetables will break up into smaller bits and end up clouding the stock.

Strain it and cool it down tonight before putting in the refrigerator.  To cool quickly, either divide amongst several pans (greater surface area is better than depth as it will dispel heat quicker) or if you don't have a BIG pot of stock, strain it and put it in an ice bath (water and ice cubes) in your plugged kitchen sink.

If you make stock a lot, you can save plastic milk jugs and fill them 3/4 with water and keep in the freezer.  After straining the stock you can place one of those inside the pot of stock to also help cool it from the inside out.

In the morning you can then skim the layer of congealed fat from the top.  If it's chicken stock, save the fat for frying potatoes with extra flavor!