I see some recipes call for you to add water to the fermentor (just before pitching yeast) to bring the contents up to 5 gallons. One recipe called for starting the boil with 6 gallons of water. Is one method better than the other assuming the water you add after the boil is purified or boiled and chilled.
One benefit I can think of is if you do it after the boil you could freeze a gallon of purified water and it would greatly aid in the cold break process.
I know of people that just use tap water. Likely tap water is free enough of bacteria that you don’t need to boil it. But you can if you like.
If you want to freeze the water to aid in cooling your wort, freeze it in multiple blocks. Larger surface areas will cool the wort faster.
Full volume boils are generally considered to be a better option than partial volume. Less chance to caramelize the wort. To avoid that with a partial boil, you can do a late extract addition.
Hop utilization is considered better with full boils. Again, late extract additions can help with this.
With full volume boils, you either need to have a chiller or cover and wait over night for the wort to chill.
Many ways to make beer. 
edit: spell check
Nighthawk makes several good points, especially his last one.
Consider intentionally shooting for less than the specified volume. That will give you a more concentrated - higher specific gravity - wort at the end of the boil. Then you can dilute it (with sterile water) to hit your desired final gravity.
I recommend this method because I consider the final gravity to be the first consideration in brewing the beer style I want to drink. The higher gravity will affect hopping level, but you can either add an extra 1/4 oz to your bittering addition or decide that the difference is not detectable (my suspicion). As you learn your system better you can decrease the intentional error until you reliably hit your desired final gravity.
