Speed:
Solid Infill & Top Solid Infill. 
Slow them down a bit. 
Usually, the infill speed is faster than the perimeters.  And that's fine for normal infill, but for the bottom and top infill slow it down.
First Layer Speed. 
Should be slower than the subsequent layers, to promote adhesion, about 75% of normal speed is a good number

Advanced:
Default extrusion
For your .40mm nozzle should be .48mm.  THis is what gives you the correct wall thickness, and dimensional accuracy.
First Layer
should be a thicker extrusion.  I use .55mm again, promotes good adhesion and a solid bottom surface
Solid infill & Top Solid Infill
I use a heavier extrusion .52mm.  This doesn't have any effect on dimensional accuracy, so you can play with it as much as you want.

Some notes:
E steps should be closer to 105
Extrusion multiplier seems high (these two balance out)
This info leads to inconsistent filament, a partial clog or a slipping extruder gear.
I am guessing your filament has an oval cross section and is having trouble feeding consistently.

looking at that last pic you posted...
Looks like outer part of it is filling/squishing good, but the center part isnt?

what surface are you printing on? the aluminum bed? or glass, or what?

Looks like you may have some bed level issues there - as in the center portion is too low.
the stock aluminum beds are known to have warping issues.

the E steps/mm seems kinda low to me...

most are around the 105 mark

mine can run as low as 103, depending on the filament used.

97 just seems awfully low

Measured extrusion was 142mm when 100 commanded at the original setting of 138, so we need to use 138*(100/142) or 97 instead.  And measured extrusion at a setting of 97 was right on 100mm.

And my calibration cube wall width was around 0.75mm before, and it's 0.48mm now, so something's right...

But the bottom looks like a sieve:

most are around the 105 mark

mine can run as low as 103, depending on the filament used.

97 just seems awfully low

And then the second layer:

Kinda looks like the perimeter is OK and the infill is way too low?

I have the E3DV6 hot-end with an 0.40 nozzle, and am printing at 240C as per E3D recommendations, bed at 100C
My settings are (defaults where not specified):

Slic3r 1.2.6
Layer height: 0.1976
First Layer Height: 100%  <---- Default 0.35
perimeters: 3
Horizontal Shells: 4 Top, 4 Bottom   <--- Default 3
Extra Perimeters if needed: Disabled <--- default enabled
Seam Position: Aligned

Infill: 100%, both patterns Rectilinear

Skirt: 1 @ 3mm, height 1, minimum 10mm of filament

Brim: No
Support: No
Speed: untouched Solidoodle defaults (not Slic3r defaults)

Advanced:
Defaults:
First Layer 200%, all others 0 for defaults.  Same thing happens when everything set to 0.48
Threads 9 (8 core CPU)

Filament Settings:
I'm using Monoprice Orange, if that matters
Diameter 1.70
Extrusion Multiplier: 0.923  (Note that this gives 0.48

Printer Settings:
Nozzle Diameter: 0.4 mm

Printer:
Steps per unit:
M92 X88.00 Y88.00 Z2268.00 E97.00  <---- Originally E138, way overextruding

Clearly I screwed something up when I rebuilt the printer and changed the stock SD head for the E3DV6, but exactly what is wrong?

Thanks All!

Looks like a pretty common bed height issue to me. You are printing too far away and/or under extruding. Adjust the z stop until the first layer makes a nice flat ribbon. What is your extrusion multiplier, filament diameter, and steps per mm on e?

If the first layer lacks squish, the following layers will also. The z stop sets the constant point from which the bed moves down, so if it's too far away to start at say c, it will still be too far away at 0.3mm + c

Actually as long as the bed is with in his layer thickness of the nozzle, the first layer will fill that gap just not be squished as much as it should. Then the next layer will actually be squished more than it should and then the third layer should be at the correct layer height and and position.

The only time you would get that layer one plus effect is if the nozzle was actually higher than his layer height and there was no squish at all. Plus it would not stick either.

If you used another layer thickness then your wall should be 60% more than that layer. So at a .2 layer your wall should be .32 or there about.

When you calibrate your bed height there is no real magic number that is good for all. Ideally your bed should be calibrated so that your first layer is roughly half of your desired layer height. Usuall around .15 to .20mm should work for most heights unless you use a .5 nozzle then you would need to move to .25 on your clearance. I use real feeler gauges when I calibrate my beds.

If the first layer lacks squish, the following layers will also. The z stop sets the constant point from which the bed moves down, so if it's too far away to start at say c, it will still be too far away at 0.3mm + c

You need to adjust your bed height.

If the next layer also looks the same I'm not sure it would be a bed height thing.