Help me troubleshoot my print results

Hehe boxes like that are a real pain for warping. Those look huge - you've done well though, should be proud!
I'd consider a few things:

Process tips:
* Reduce layer height: try 0.3mm, or even 0.2mm instead of 0.4mm. this will improve interlayer adhesion.
* Increase ABS extrusion temp: 210 is typically a bit low for ABS in an E3D, try 230. This will improve interlayer adhesion.
* You might want to calibrate the steps-per-mm in the extruder with the PLA
* Wall thickness calibration will be handy either way - looks like in the PLA you're definitely overextruding, but in the ABS it's hard to tell. Lawsy carriage print in black looks a bit fat, but there's something else going on in the base of your white box that I'm not sure I have a clear answer to just yet.
* For ABS, ambient temps as high as you can get; aim for 50-55C. Even consider going as far as adding a heater to the enclosure. This will reduce the thermal forces, which is the root cause of both warping and layer splits!
* Check how much z backlash you have. If you can get a dial gauge and measure it, that would be ideal, but you can usually work it out by printing a calibration pyramid (the 5mm cube pyramid thing, must be on thingiverse somewhere), and measuring the difference between the height of the bottom layer and the subsequent layers - and then rather than get all mechanical and install crazy ballscrews, just make sure you have the community firmware installed and use M99 to compensate. This will help get the first few layers right (I can see evidence of some catchup happening in the lawsy carriage, but this could also be due to just overextrusion).

Design tips:
* Consider rounding the corners of the box in the design - this can reduce the propensity to lift
* Split the box in the middle rather than at the top so it's two halves bolting together, rather than a box and a lid. This way both parts share the pain
* If you can use PLA, use PLA. Be wary of anything heat-generating, as PLA has the lower melting & glass transition points.
* As a last ditch, consider splitting the box into individual walls and floor with pins to align the joints, and then glue the thing together. Perhaps it's a bit audacious. This will also help you get neater cutouts etc. as they would be printed horizontal.