it occurred to me that Root calls the Machine 'she' and is also the only one actually directly talking to it. Do you think the Machine asked her to use that pronoun? Could that make the Machine somewhat bigender, given Ernest Thornhill? I think that would be kinda awesome.

Honestly, I don't think the Machine genders itself. Root canonically likes to project her opinions onto others (like when she had Harold captive) and also explicitly thinks the Machine is God, which is something I highly doubt it agrees with. I think the Machine is just okay with letting her believe what she likes as long as she will work for it without killing people. (perhaps it picked up Harold's manipulative streak, eh?)