Yes, per capita literally does mean "per person", but the purpose it serves here is to allow meaningful comparison of more than just raw numbers. Manhattan might have 1,000,000 fixies, and Brooklyn might only have 100,000 fixies. This by itself doesn't really tell you much, since what if Manhattan had 1,000,000 people and Brooklyn only had 50,000 (again completely theoretical); then, the "per capita" fixie bike index would be twice as high in Brooklyn.
I don't get your point with the example, for those values I get "fixies per capita" of 1 in Manhattan, and 2 in Brooklyn. That clearly states that there are twice as many fixies per person in Brooklyn as there are in Manhattan.
If you also normalize based on Manhattan, you'd get 100 and 200 which say exactly the same thing but with more zeroes and no longer being "per capita".
Yes they are. The per capita value is meaningful, albeit inaccurate, because you're estimating the value based upon a sample.
Whatever number you multiplied every single index to get it into the range [0, 100], divide by that to get the per capita value.