Okey dokey. Out of these three breeds, which would you consider the tamest?
1. Black Australorp
2. Barred Rock
3. New Hampshire
Also, which would you consider the best egg layer, or are they all equal?
I believe in every breed there are roosters who desire to be tame and/or given attention, and those who don't.
That said, the tamest, most docile roosters always seem like the happiest males to me when we interact with the flock. Some males can be extremely well balanced. Calm, unafraid, less skittish, and friendly with humans, good with other males, and often interact well with other creatures (dogs, cats, horses, etc.) and amazing with hens, or juvies.
My votes are 1, 3 then 2 for the best chances for more docile males. BR in the hatchery form have been occasionally been said to have a bit more in the aggressive male side- however, there's always exceptions though in any breed, good or bad- especially in hatchery birds when there is no selection for aggression, so it's a lottery. All are very good producers of eggs. We used McMurray when we purchased our first hundred birds, and only had one extra ornamental chick pass, the rest were pretty hardy heathwise- although many suffer from brittle feathers, small birds, comb sprigs, etc. Although we had PRIR (many said to be aggressive) and a Partridge Cochin, none of the 18 original males (straight run) were aggressive.
However, the overall best LF breed we've had personally are Black Copper Marans. Both hen and rooster are very calm and affectionate and don't mind handling if they trust you. They have everything going for them. Large, very attractive birds, with beautifully colored, silky and pliable feathering. Good representatives of the breed can be shown competitively, and the meat is considered among the very finest in flavor. They have gorgeous dark brown eggs (some hens are better layers or darker layers than others) and usually the egg is very round at center- most often, at least a L, up to about 3X much of the time. (Less smalls and meds than other breeds we've had.)
They are not generally known overall to lay as much as production breeds, however, some hens have been known to keep up a decent pace for a good period without slowing, and many seem to lay well into over 2 years old, unlike many breeds. Ours laid well into winter with only a 3 week break. We have one hen about 3+ years old who lays huge 3X eggs, generally 4X weekly. Another hen lays in spurts of 10 days straight before stopping for a few days, and re-cycling the paint color- she's a better layer than our PRIR. However, if you ever decide to purchase some, purchase only from a reputable breeder specializing in BCM. Marans from hatcheries have not shown good results- egg color plus conformation can be tricky. Just too young of a breed in the US to produce well commercially as of yet.
Good luck!!!