Technically, the first decision is Porter's. He gets to decide whether he signs an offer sheet, and if so, which offer sheet he signs (if there are multiple offer sheets, which seems likely). If Porter decides to sign an offer sheet, the only remaining decision lies with the Wizards, who decide whether to match the sheet and keep Porter, or to not match the sheet and lose Porter to the team whose offer sheet he signed.
Porter has to make the decision of which offer sheet to sign before the Wizards decide to match. Thus, if Porter signs an offer sheet and then the Wizards decline, Porter must go to the team whose offer sheet he signed, not any of the other teams. The decision gets winnowed down to two possible teams when Porter signs an offer sheet.
Once Porter signs an offer sheet, then the sign-and-trade is no longer available. He will either go to the Wizards or to the offering team, in either case under the terms of the offer sheet he signed. However, if Porter DOES NOT sign any offer sheet, the Wizards can negotiate a sign-and-trade with any legal team (including teams that did not submit an offer sheet). Typically, the terms of any such sign-and-trade would be identical to the offer sheet Porter is considering signing, and the sign-and-trade would be with the team whose offer sheet Porter intends to sign (because Porter must consent to the sign-and-trade in the same way he would consent to any other free agent contract).
Porter and the offering team have no requirement to enter into a sign-and-trade negotiation with the Wizards. Porter can simply sign the offer sheet and end any consideration of a sign-and-trade. If Porter selected the Kings' offer sheet, and the Kings wanted to play hard ball, they would likely ask Porter to sign the offer sheet and hope that the Wizards did not match. However, if the Kings think that the Wizards might actually match, then they might be interested in negotiating a sign-and-trade, being willing to give up some assets in order to take away the possibility that Washington matches.
My personal preference (if Porter selects the Kings) would be for the Kings to play hard ball, unless the Wizards offer a sign-and-trade for minimal return (say Koufos, or a second round pick or something). Otherwise, have Porter sign the offer sheet and force the Wizards to match. I think with their salary cap situation there is a good chance they will not. If they really wanted Porter on a deal of the max size that other teams can offer, they'd have just put that deal in front of him on July 1 and ended all the drama. (If they offer that contract, they're obviously willing to match the same contract, so why go searching for the offer sheet?) So I tend to think the Wizards are bluffing, and we should call it.