In MTG: Is it true that a card can't tap unless it has been in your control since the start of your turn?

This one can be tricky. The immediate answer is no, Xbox arcade was not wrong and I have never had a problem with the rules there. The detailed answer is as follows: The reason that certain cards cannot attack or use tap abilities the turn they come out is because of summoning sickness. You may have notice that this only applies to creature cards, which is true. Creatures come into play with Summong Sickness, and this means that they cannot tap or attack until your next turn. This can confuse many players by thinking that you may use a tap ability on your opponents turn after you play a creature. This is not true, you must wait until it becomes your turn again. Some creatures have haste, which allows them to attack or use tap abilities the turn they are played, and certain spells can give creatures haste. All other permanents can be tapped, including artifacts, the turn they come out unless otherwise stated on the card. I hope this helped :)