Given the importance of sanitation, particularly after the boil, should I boil water before adding it to the cooled wort to bring the volume up to 5 Gal? Tap water is of good quality and tastes good.
I have no idea about the OP's water but here in Sussex UK I use the water straight out of the tap. I would not boil it - I would just add it to the wort in the FV, pitch the yeast and let it ferment. In most situations, fermentation is quite capable of dealing with and neutralising most common (and virulent) pathogens. Indeed, it is one of the useful effects of brewing and was used for rendering water safe to drink in centuries past. The real "Ginger beer plant" is especially good at this.
I go by the maxim that if your tap water is safe to drink then it is safe to brew with.
Of course there is nothing wrong with boiling the water and allowing it too cool before adding to the wort. It just increases the "carbon footprint" of one's beer production.
It depends, obviously. In many countries tap water is close to a few microorganisms in a milliliter, which would theoretically guarantee it to be safe. There should be test results available online or you could get your water tested in a lab (depending on where you live). However, if you are worried, it's better to boil it, just for your peace of mind.
For the future, it would be a good idea to make a simple test - pour some of your normally sanitized wort to two containers, add some boiled water to one of them and some un-boiled to the other. If the container with not boiled water spoils faster you will know that you need to sanitize your tap water.
Personally, I would wager that microbes that you missed during normal sanitation (and you can be sure there is plenty of them, no matter how careful you are) outnumber those in the water greatly.
I have well water which regularly tested but which also passes through a charcoal whole house filter. As a result my water is very safe and has low microorganism content so I do not boil. I would however if I lived in a city and my tap water did not run through any filtration system. A simpler solution might be to install a tap level water purification system on your tap of choice in your brew day area and use that water. They are not expensive, are easy to maintain and install and work very well. Just a thought....