Water – Code requirements for the proximity of Power outlets to wash machine water outlets

Yes, it's probably not overly difficult.

Water lines

Water lines are pretty easy, just extend them over or run them from a new location. Don't leave dead-legs (or dead-ends), so depending on where you're coming from you should cut and cap the lines as close to the supply as possible, eg 1" away. Dead-legs don't have much water movement, so can become harbours for bacteria which will then get into the rest of the plumbing system.

Waste lines

With waste/drain lines, there are a couple more concerns. You need to move both the vent line and drain line over separately, as vents must be connected within 5 or 6' (depending on 1 1/4 or 1 1/2" line, respectively) of the trap.

You also need to ensure all the waste pipes run downhill towards the drain, and there are no places for water to pool (except for in the trap, of course).

Dryer vent

You want to minimize the distance the dryer vent goes, to minimize the resistance. The less resistance, the harder the air will blow, and the less chances for lint to get stuck and clog it up (which can lead to a fire). The best course of action may be to close up the old hole and make a new one, close as possible to the dryer. I prefer to use 4" solid smooth duct lines whenever possible, and the semi-rigid lines after that in walls, if needed.

Don't put the tin-foil-like lines in the wall (it's okay for the final hookup between the dryer and the wall, but that's it), and don't use the white plastic junk whatsoever.

Electrical

You'll need a dedicated 220V circuit for the dryer, and then another circuit for the washer and plugs in the room. You can probably extend the existing dryer outlet by converting the existing receptacle into a junction (with blank faceplate) but of course it depends on what you are doing with that space after. All junction points must be accessible.

NEC 210.50(C) states the washer outlets must be GFCI-protected if within 6' of the top of the laundry tub. In my laundry room, there's one circuit for a couple outlets including the washer -- the first one is a GFCI outlet, and the rest are connected on the load side so they're all GFCI-protected. I'd recommend just doing something similar, it's not at all difficult and much safer if there's ever water on the floor, etc.

Floor Drain

I don't believe there are any specific code requirements for floor drains for a laundry room, but there may be specific requirements in your local building code (check with your planning office). Either way, it's something you should think about. Washer lines and washer valves do sometimes leak, and can cause a fairly big mess. Having a floor drain in the room will help minimize the damage if this ever happens. It may not be worthwhile to dig up the floor to put one in there, but at least be aware of the path the water would have to take to get to the drain (eg, does it have to go out into another finished room, across the carpet, and turn a couple corners before it gets to the drain?).

Insulation

Since these lines are often installed on an exterior wall, you'll want to think about insulation for them. The picture above is actually from my laundry room. I did spray-foam insulation on the basement walls, and though you can't see in that picture, there is a gap between the framing and the walls: most of the basement it's about 1", but on that section it is closer to 2", just to be sure the pipes are not in the cold zone.

Yes, there are

The Code cite you're after is NEC 210.52(E) (with subsections 1 and 3 being the commonly applicable ones):

(E) Outdoor Outlets. Outdoor receptacle outlets shall be installed in accordance with 210.52(E)(1) through (E)(3).

Informational Note: See 210.8(A)(3).

(1) One-Family and Two-Family Dwellings. For a one-family dwelling and each unit of a two-family dwelling that is at grade level, at least one receptacle outlet readily accessible from grade and not more than 2.0 m (6 1/2 ft) above grade level shall be installed at the front and back of the dwelling.

(2) Multifamily Dwellings. For each dwelling unit of a multifamily dwelling where the dwelling unit is located at grade level and provided with individual exterior entrance/egress, at least one receptacle outlet readily accessible from grade and not more than 2.0 m (6 1/2 ft) above grade level shall be installed.

(3) Balconies, Decks, and Porches. Balconies, decks, and porches that are attached to the dwelling unit and are accessible from inside the dwelling unit shall have at least one receptacle outlet accessible from the balcony, deck, or porch. The receptacle outlet shall not be located more than 2.0 m (6 1/2 ft) above the balcony, deck, or porch walking surface.

In other words, there needs to be a readily accessible receptacle not more than 6.5' above the walking level in front, in back, and for each balcony, deck, or porch you have.