EX13-035 KingEtemon

If activated, first choose if you are going to return 10 specified cards from your trash to the bottom of your deck. If you do, this effect's play cost maximum is increased by 6.Afterwards, you can play up to 6 total play cost worth of specified cards from your hand or trash without paying the costs.

No, you can't.All of the conditions within "by doing X" must be fulfilled.The condition for this [On Play] [When Digivolving] effect requires you to return 10 specified cards to the bottom of the deck.

Yes, it references your opponent's Digimon as well.