In nullsec (0.0 security space) organizations claim/seize/otherwise acquire space and establish sovereignty (sov) over it by somewhat more formal and complex means. In lowsec (low security space) space cannot formally be held, but organizations claim areas as their “turf” and as in nullsec, if they defend it then it is theirs. In hisec (high security space) the four empires, glorious Amarr, honorable Caldari, gallente, and (*spit) minmatar degenerates own all the space.
Yes, of course…usually…there are exceptions.
Yes…but many corps do many things. You can certainly be a specialist in one area for a corp (especially a larger one)