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:
- 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.
- 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.
Well, your pictures have twigged an occasional issue I run into where I can't see them, (likely not your fault) but flying blind....
From a functional point of view you really don't have to worry about separation. Twisted pair is actually quite good at ignoring noise, and 60 Hz noise is of little note to 100MHz ethernet anyway. You can do it all wrong and it will work, 99% of the time. I don't suggest that you do it all wrong, I do suggest that you don't freak out about it.
From a code (and safety) point of view you should not have low and line voltage going through the same hole, and from a hyper-cautious network standpoint they should be separated by a good 12" when parallel, or cross at 90 degrees if they need to cross. Separation matters a lot more in an industrial environment with noisy devices on the powerlines than in a typical single-family residence.
From a "best practices" point of view, network and power in separate stud bays (when running vertical) or between different sets of joists when running parallel to the joists is certainly a best practice, though not required by code - it maintains separation quite aggressively.
Notches are far worse than "holes in the center third" (top to bottom) of a beam, joist, or header. Holes should not be too close to the ends, even in that center third. I'm not sure there's any need for the cables to go through the header if they can "float by" (remember, I'm flying blind here)
Portable electric heaters are nasty, nasty loads. If you are plugging one into a 15 amp (14 Ga) circuit that, all by itself, with nothing else on, is loading the circuit very near to maximum (if it's a 1500w heater, actually more than is permitted for fixed/hardwired loads that might (as with a heater) be on for more than 3 hours. A warm wire is fully expected under those conditions. A 20 amp (12Ga) circuit would be more appropriate if you are in the habit of using one.
Best Answer
In my jurisdiction they would call that bundling and flag it. They normally won’t say anything with 2 cables and some areas up to 4 but when there are more than 3 current carrying we have to start derating
over 8 current carrying conductors the built in safety in #14, #12 
 exceed what is allowed.
there can be some issues there even though nipples less than 24” don’t have to be derated a long run like in the photo is normally flagged with more than 4 is all the states I have worked in.
I see 6-3 , 8-3 & 12-2 x2 10 current carrying conductors as I said most jurisdictions would flag this as bundling.