The text reads "Action: Choose a Silvan or Noldor ally you control. That ally gets +2 until the end of the phase. At the end of the phase, shuffle that ally into its owner's deck if it is still in play."

Can this be played on objective ally Arwen in The Redhorn Gate scenario? If so, do we lose the game at the end of the phase, since arwen would "leave play"? Or is that part of the text ignored since there is no "owner"?

I think the encounter deck owns objective allies:

The scenario is considered to be the owner of the encounter deck and each encounter card.

[RR Ownership and Control , http://www.lotr-lcg-quest-companion.gamersdungeon.net/#Rule562 ]

So you should be able to play Children of the Sea, but you'll lose at the end of the phase (if you didn't manage to win beforehand).