99% of the time, "This is John's school" means this is where John attends or attended school. (Or possibly teaches.) (If John were the founder of a school, it could mean "This is the school that John founded.)
The majority of my club's new English learners often generalize B of A = A's B or vice versa. The formulae do not always work as Grammar Geek pointed out. I use them sparingly unless I sense that B is a member of A set, such as: the cadets of the school = the school's cadets,or the windows of the house =
According to WordNet - bird's nest, bird nest, and birdnest = nest where birds lay their eggs and hatch their young. The phrases, however, by themselves, do not say whether the nest is human made or bird made; additional context is required. Moreover, as you might already know, people also talk about bird's nest soup, which is made from the saliva of birds / bird