Electrical – What’s the appropriate cable type along exposed rafters

Do not use lamp cord to extend a circuit to a new outlet! The wire is neither big enough, insulated enough, or grounded. It's illegal everywhere, and I wouldn't consider it to be safe.

Use Wiremold with standard electrical wire. That's why the product exists.

The crawl space case

Wiring in a crawl space (wet location) can be done successfully using a variety of means:

• THWN (pretty much all building wire you buy at the borgs these days is both THHN and THWN) in either rigid metal conduit or schedule 40/80 PVC conduit
• Type UF or NMC multiconductor cables, but not ordinary NM-B, or
• Type MC cables rated for wet locations (i.e. those with an overall PVC jacket and THWN or XHHW-2 internal wires)

However, given the circumstances (i.e. the persistent wetness in the crawlspace), I'd be most comfortable with THWN in PVC and nonmetallic junction boxes down there, with the risers being THWN-in-PVC as well. The main downside, though, is that not only is this an expensive way to wire a house, most "wet location" electrical equipment will not survive being bathed in another storm surge!

Wiring in soffits

The soffit proposal is elegant; however, the main problem is going to be providing access to junction boxes -- you'll need to have blank plates interrupting the molding (preferably on the bottom of the soffit) in order to provide access to junctions:

314.29 Boxes, Conduit Bodies, and Handhole Enclosures to Be Accessible. Boxes, conduit bodies, and handhole enclosures shall be installed so that the wiring contained in them can be rendered accessible without removing any part of the building or structure or, in underground circuits, without excavating sidewalks, paving, earth, or other substance that is to be used to establish the finished grade.

Further more, you'd have to use an "old work" type of junction box in this application as you wouldn't have framing members inside the soffit to attach the boxes to. While the 3" by 3" soffit interior should be adequate for cabling, I would not use a conduit inside it due to the soffit interfering with conduit body access. I do not know of any fire code requirement for horizontal cable chases other than them requiring firestopping when they penetrate a firewall; however, as mfarver pointed out, your AHJ has the final say on that, and internal firebreaks in the chase would be wise.

Could a busway be the best way?

Another option for the soffit wiring case would be to use a plug-in, non-ventilated, totally enclosed busway system with branch circuit breakers at the outlet taps and type MC cable run exposed or NM run in surface raceways for the drops/horizontal runs to outlets and luminaires. While unusual for a residential application, plug-in busways offer a high degree of flexibility in layout, and can be subdivided so that branch circuits can be moved with only modest impacts on power to other parts of the house.

There are two drawbacks to this approach, other than it being relatively costly, though:

1. A means of access into the soffit that does not damage the soffit would need to be provided: either the soffit side panel could have hinged access panels in it, or the screw system designed so that the side panel can be removed and replaced without causing any damage to it. See NEC 368.10(B) for details.
2. You'd need to locate the branch circuit breakers at the tap-off points -- while this doesn't limit the height of the busway, as per NEC 368.17(C) and Point 1 in 240.21(A), there has to be some sort of rod, chain, or what-have-you attached to the breaker handle so that it can be operated from floor level.