In example A you cannot tap to add a creature to the opponents side because there isn't a point in the untap step where you can do anything. You go to untap step permanents and such untap then you're immediately in the upkeep step at that point Oath of Druids has already checked and saw that the creature count is 1:1.
Not quite. It has checked the creature counts in that it has had it's target assigned but the ability goes on the stack and thus anyone can respond to it.
Im not sure this is how it works. Maybe im misunderstanding your point, but the ability won't go on the stack because of what Hagrid mentioned, right? You cant respond to the trigger if its not on the stack, and it won't be on the stack if there isn't a legal target for Oath at the very instance of the start of your upkeep. There won't be a legal target because there is no point at which you can change the number of creatures the opponent has before the start of your upkeep
Neither player receives priority in the untap step (abilities that trigger when a permanent is untapped carry over to upkeep).
At the beginning of the upkeep, Oath of Druids triggers. The controller of Oath of Druids is then required to pick a legal target. If they cannot do so, the ability cannot be placed on the stack and "fizzles". If they are able to do so, the trigger is placed on the stack. After other upkeep triggers are placed on the stack, the active player receives priority and this is the earliest point they would be able to tap Orchard.
The ability will check when it is about to resolve whether or not the target remains legal. If the target is now illegal (opponent is untargetable or creatures are now equal), the ability will be "countered on resolution", just like every other spell and ability without a legal target. If the target remains legal, the player chooses whether or not he or she would like to use the ability as it is a may ability. Cards are milled, creature is put into play (or whatever), and then the active player gets priority, yada yada yada.
The reason for the somewhat convoluted wording is that the player who controls Oath of Druids is the controller of the ability. "Target opponent" refers to an opponent of the controller and so it cannot be used for the wording on Oath. Why is this relevant? Well, it is the controller's responsibility to remember their triggers and they must trigger Oath on their opponent's turn if the conditions are met. If they forget, it should be a game-rule violation and the opponent should have the ability to put it on the stack like any other detrimental trigger. If they intentionally do not put the trigger on the stack, this is blatant cheating and the punishment could include a disqualification. This is of course at the judge's discretion.
Hopefully, this was an adequate summary.