I'm not sure I'm understanding either of your questions correctly.
For the first case, if a unit is Kidnapped, and then leaves play and comes back again (due to e.g. Geiger or Prynn maxband, or Second Chances), then it remains under the Kidnapper's control.
If it's a Rambassa Twin, then it will try and fetch another twin from the Kidnapper's Codex. If they don't have a Twin in their Codex, the 'Arrives:' ability will fizzle. If they do, then it will fetch one. At this point, you need to keep track of which Twin is which, because they'll go to different discard/hand/codexes when they leave play. A Rambassa Twin which is arriving while under Player A's control will never summon a Twin from Player B's codex. A similar case will occur if you somehow manage to get an opponents Rambassa Twin in your Graveyard (though e.g. Kidnapping a Twin with Graveyard in play, and trading Twins)
For Kidnapping a BRO, 'End of Turn:' abilities will always resolve before effects that last 'until end of turn' end. So in this case, there is no choice to make: if you Kidnap a BRO, and don't attack with it, it will go back to it's owner's hand before it returns to its owner's control.