Here you can just see one 'switch' wire to simulate a TRUE into 1. and FALSE into 2., however, the output is still on. I have connected two actual switches up but still the output is on.
Question 1: The output seems to be High all the time regardless of input. Should the IC be giving High output when no switch wires are even attached?
Question 2: Could it be that the IC is simply not working or that I've wired the circuit incorrectly?
You misunderstood my comment, which I see was ambiguous. I was agreeing with you. I meant that, as you stated, an unconnected goes to HIGH or TRUE. It does not go to LOW or FALSE as the OP apparently assumed, since he tied one input to HIGH and left the other input open.