If the connections skill raises your personal effective standings with a faction or corporation, does the connections skill also raise your corp's standings with a faction or corp, like for use with anchoring a tower, if you're the only person in the corp?
No, player corporation standings derive from a straight average of the standing all characters in the corporation to a unique entity. If a character does not have standing to that entity, their standing is not taken into account.
e.g. if a corp has 4 characters in it, and 3 of them have standing to Caldari Navy of 4.00, 5.00 and 6.00, and the fourth has never run Caldari Navy missions at all, the corp average would be ( 4 + 5 + 6 )/3 = 5.00 ( the fourth character with no standing is not counted at all in the equation). This still applies if there are no other characters in the corporation.