Yep, all quite good points Dick and hashie.
Chaps, I'm adding all this below for the OP, can consider for an article.
With in- kettle cooling (be it passive by just leaving it to air cool, or enhanced by water- cooling in the laundry tub etc.), airborne pathogens are a possibility in just leaving the kettle open, particularly where there's any grain dust or flying insects.
With a stockpot though, the lid is an excellent fit, so that's what I use. I've seen some folks put tape over the breather opening in it, but most of the time I'm happy to leave it open while the wort cools and also contracts. Not everyone can put a well- fitting lid on their kettle though and while foil isn't the most reliable way to keep airborne gunk out, that's about all some folks can do.
No- Chilling takes advantage of the heat from near- boiling wort to pasteurise the internal surfaces of the cube and its contents, when hermetically sealed by putting the lid on, then the wort cools either passively in air or enhanced by 'slow- chill' and putting it in a swimming pool or tub of cool water. The wort contracts as it cools and sucks the sides of the cube in, this is normal, while the obvious sign of an infected cube will be bulging a few days after filling (quite rare!). Brewers should check that the cube is made of HDPE (class 6?), which is usually considered 'food safe', likewise if the cube has previously contained some other materials then ensure that it is well- cleaned and rinsed (hot) with no residues remaining.
I've stored No- Chill cubes for weeks, while I've heard of folks leaving them for months. I'm fairly sure that Fresh Wort Kits (FWKs) are made up as No- Chill cubes and transported all around the country and beyond with very few problems.
With NC cubes, to be honest, I've never been overly concerned about excess headspace and will sometimes just half- fill a cube, however in that case I'll make sure the hot wort gets to all surfaces for a few minutes to ensure good sanitation. Brewers can minimise the headspace by pressing the sides in to expel all of the air before sealing. I always fit a tap to NC cubes, it helps when expelling the air and also at emptying, however it does need to be checked thoroughly during the cooling that the spindle doesn't pop out and also that the tap thread is correctly seated, also remember the tap is quite vulnerable to being knocked or bumped off as the thread is less likely to resist any abuse while heated.
I've even pitched directly into a NC cube of wort, but it needs some preparation in fitting the tap beforehand (although it can be fitted just before pitching too). Fill the cube as usual, when ready to pitch, take the lid off to release the suction and let some fresh air in, pour in the yeast starter culture and re- cap, shake it up a little to aerate, re- open and cling film over the opening and you're done- it couldn't be simpler! Also this introduces another use for the cube as a fermenter.
There is a way of letting the cube cool with the break material around the tap for removal before pitching, but I've only tried it once and it was just too wasteful to be effective. I'll report back when I have another go at that.
Otherwise, when I'm emptying a cube into a fermenter, we'll note that most of the break material and debris has settled to the bottom of the cube during cooling. Now is a good time to separate the good clean wort from that muck, so I'll gently run the wort out through the tap, trying not to disturb the layer at the bottom. A cube standing upright will have about 1.5 to 2.0L of 'deadspace' when the tap is open, so just leave it all in there!
I probably break a few rules with kettle- chilling and NC cubing, but that's just how I operate, when it come time to write up the article, we might care to exercise some editorial discretion! Plus I have written yet another verbose essay...