If they can't be trusted to not lose the card, yes, that's probably a good idea. There will likely be times when the kids may be by themselves -having the card on them then would be a good idea.
While Plusgrade handles most of the Royal Up process, the initial bidding still happens within RCI's control - it's RCI that provides the deceiving pics.
Realize that whatever answer JC gives you today can change by the time you sail - heck, the answer could vary on the same sailing between different bars.
And yet, on Q class there typically aren't any reservations available for the show in the theater and, there's usually no chaos. You are incorrectly assuming shows are that popular and everyone will head there at the same time - it just doesn't usually happen. Pretty sure there's no chaos going into Cats.
Don't ever book a cabin based on one of the perks listed on the RCI site - yes, it is normal for the web site to be out of date or something has changed and RCI no longer provides that perk.