Or, if true, could there have been a deliberate effort by staff to set the president up to look bad?
As for the bowing, Americans don't bow to foreign monarchs ... well, or didn't until recently. And I have read several times that bowing and curtsying by her own subjects is not required. I think the ‘Trump went first’ bit is cultural. How many times as host have you seen Trump defer to his guests to precede him ... maybe even with a hand on their shoulder? As guest and equal, I'd expect him to lead.
I suspect that when Trump walked in front of the Queen to review the troops it was just a miscommunication. He would not be deliberately rude like that. She is so short that her mouth is pretty far away from his ear. Combine that fact with the factor of the accent and it’s easy to see how a miscue could occur.
When they were descending and ascending those stairs the husband and I both noticed that DJT was wanting to offer an arm to the Queen. You could see it in his body language. But I suppose royal protocol does not allow that.