1. Yes, you wired the NEMA 14-30 receptacle correctly. That receptacle has two hot connections, each 120V but on different phases, resulting in a 240V potential. So if you take a voltage tester you will find it reads 240V between the two rectangular slots on the outlet, and 120V between either rectangle and the neutral or ground. You can check this using your voltage tester, but it sounds like you did it correctly.

2. Figuring out where that wire went is going to be tricky. If you can't find it visually or by process of elimination, the best tool to help would be a tone & probe kit. You'd hook up the tone generator to your mystery wire (after double-checking that it's not live) and take the probe component with you. When the probe is near the toned wire, it will beep, even if it's a few inches away inside a wall.

According to the National Electrical Code, yes it should be a 20 ampere branch circuit. This circuit should also not have any other outlets, other than the ones for "laundry equipment".

So keep the 15 ampere circuit for your general use receptacles, and run a new 20 ampere circuit for the "laundry equipment".

Also note that the receptacles in the laundry area must be GFCI and AFCI protected (210.8(A)(10), 210.12(A)), as well as tamper resistant (406.12(A)).

National Electrical Code 2014

Chapter 2 Wiring and Protection

Article 210 Branch Circuits.

210.11 Branch Circuits Required.

(C) Dwelling Units.

(2) Laundry Branch Circuits. In addition to the number of branch circuits required by other parts of this section, at least one additional 20-ampere branch circuit shall be provided to supply the laundry outlet(s) required by 210.52(F). This circuit shall have no other outlets.

210.52 Dwelling Unit Receptacle Outlets.

(F) Laundry Areas. In dwelling units, at least one receptacle outlet shall be installed in areas designated for the installation of laundry equipment.