My own reading is that there is no requirement in the Constitution for the the taking of the oath or affirmation to be in public. If there are witnesses to the oath or affirmation then it stands.
There have in fact been private unrecorded swearing ins on the death of a sitting president so it’s not completly unprecedented.
Yes the President would be able to issue orders to deploy. “Who would decide that and how?” is probably the wrong question - it’s really “How could someone challenge that?”
I don’t have strong views on the answer but impeachment and conviction in Congress is the most obvious. I doubt that the use of Quo Warrento is constitutional (but I could be wrong).
And if there are no witnesses to the oath?