Yes, they definitely do act differently.
When the eggs hatch, the hen will want to keep all other chickens safely away from her babies. I've seen hens that insisted on 10 feet of space in all directions (this is a problem in small coops!)
While she is broody, the hen mostly sits on the nest and ignores the rest of the flock, who mostly ignore her unless they want to lay eggs in her nest. But when she does come off to eat, she practices the "leave me alone" behavior that she will use after the chicks are hatched. The other chickens mostly respect that, although not always (they may say, "you belong at the bottom of the pecking order. I won't tolerate you acting like that!")
Nope. People have been deliberately breeding less-broody and non-broody hens for a long time.
Some hens go broody frequently, some rarely, some never.
I've had one hen that laid 13 eggs and went broody. After hatching & raising chicks, she laid 13 eggs and went broody. She basically did that all summer long-- great if you want to hatch lots of chicks, but awful if you want eggs.
I've had some hens that would go broody in the spring, once each year, and not the rest of the time.
And I've had ones that never went broody.
There are some breeds that are more prone to going broody than others, but breed is never a guarantee (some hens just do not read the breed description!)
In general, the hens that lay the most eggs per year are the least likely to go broody (because broodies are not laying.)