Yes, a neutered male dog will not have as strong a scent and there won't be as much testosterone in the marking urine.
They can as they still have instinct even if they have been neutered.
Not all the dogs and cats that are adopted from a shelter are always neutered.
Other dogs butts
They look and act differently
no
Yes. Male dogs will recognize pups as such and will generally behave differently than they would toward an adult dog.
To prevent them from siring more puppies.
Yes, greyhound rescue organizations neuter their dogs before adopting them out. Sometimes the dogs arrive at the rescue intact but they are neutered before leaving.
Yes. Neutered males remain fertile for up to two weeks following the procedure.
No, dogs smell with their noses.
It is a natural dog behavior to smell "that area" on other dogs and people, because this is how dogs gain information about the person/dog that they're smelling. Much like people get to know other people by what their faces look like, dogs tell people (and other dogs) apart by their smell, and the part of the body where that individual smell is strongest is, well, the groin/rear. So, it is just how the dog tells you apart from other people.
You can have a neutered male and female boxer without worrying that they will try to mate. The dogs will not mate if the male is neutered.