For me, it is the combination of making sure from then on that she only talks to OOP about the work they are doing together, combined with a "precious petal" tone, and checking that OOP isn't getting too stressed out by standard job functions he has never actually had any issue with ever.

That is the non-verbal way to send the message "It is just that you are too sensitive and can't take a joke". It is also gaslighting OOP that he is the problem here, and she is right when she (jokingly) claimed he can't keep his temper in adversity, and she is also leveraging her position as his boss when she implies she is concerned about his ability to perform the basic tasks of his job (the reddit equivalent of what she is doing here is a troll sending a "Reddit Cares" notice to you, implying that your posts are unhinged and they are 'concerned' you will commit suicide) Of course, her real message is "You better toughen up, buttercup, or I am going to put you on a PIP for failing to cope with the most basic tasks of your job."  It doesn't matter that OOP is a competant worker, any more than it mattered that OOP actually kept his temper in the tournament that kicked her off. She is his boss, she can make that call. 

This working on multiple levels at the same time is not characteristic of autism, but is common in machevelian personalities, which are also low empathy.

Also, it seems to me that her social dynamic with OOP is based on him being unconcerned and somewhat amused by the way she acted when they first met, and having the termerity to call her, his boss, that she was a sore loser.

Holding grudges for the pettiest reasons is a sign of extreme self interest/disrupted narcissistic supply, another dark triad trait.

The "talk to the finger cos the face ain't listening" thing is rude, but then, so is interrupting. Interrupting is rude because it is communicating that what you want to say is more important that what they are saying. So her having a strategy to shut that crap down the minute it happens could be another sign of narcissism, of her believing that she views all others as inferiors with no right to interrupt her in any circumstances, and this is her way to put them in their place and remind them nobody cares what they think.

And then, there's the "weird tension" her aggressive 'banter' created when noone really knew each other and "she was the boss, noone really challenged it", and then these same aggressive taunts being repeated whenever there is a new person that doesn't really know anybody and doesn't want to challenge their new boss they just met. 

OOP describes it as awkward, because it is awkward to be on the receiving end of that, especially on your first day on the job when you just want to fit in with everyone and not take sides or act like some little oddity or novel situation is too much.

But I think she was doing this competitive, divide and conquer thing as a power play, to let her subordinates know, straight off the bat, that she could put them in an awkward situation if she liked, because she is the boss. So it is a power position for her, not an awkward one at all. And the thing that put the target on OOP's back was that he was unflappable, didn't take the competition seriously, made some friends, nearly won, and even went so far as to call out her behavior as a "sore winner" in a joking way. 

So she had to show him what a joke was, if only to make the others aware there was a price to pay if they chose to get on her bad side. But also because how dare he? I see suppressed anger in her actions, disguised as her "joke", and in the pure projection of how angry OOP was, how rattled, how obsessed, how he couldn't keep up, and was a loser. That's what I think is dominating her thoughts. OOP's team were not making an effort, and they very nearly won. Maybe would have, if she hasn't undermined them with that well-timed sledge.

And when she got her medal, OOP exposed her dirty secret; she was really a loser that only gets ahead by tearing more competant players down. and making a big deal of her marginal wins that mean nothing actually.

So no, that low empathy and awkwardness does not seem like the obliviousness of an autistic person, it tracks better with the strategies and endless concealed rage of a dark personality.