Thanks for your interest! I'll try to answer all the questions to the best of my ability in one email.
Data: They will definitely be uploading all of the data as soon as it's released (which means tomorrow, plus a week or so to shift it from the "legacy" format to the current layout). Princeton has a lot of computing power.
Local redistricting (county/city/etc.): I don't think that they will be able to do local governments, unfortunately. This is a nationwide system so they will already be trying to respond to 150 different map-making exercises (one congressional, one state house, and one state senate for each state). Given multiple maps for most of these, that's a lot to take on.
Compactness/other criteria: They will be providing standard metrics, as I understand it. I don't know how much capacity they have to track each state's specific systems, but they may be able to add something on request. Can't speak for them on that, though. Local analysts may also be able to add that.
The main thing that we as activists need to do on the front end is to put pressure on the legislature (or other map-drawing authority) to make the shapefiles public for each map as it is released. These are the files that enable a computer to read and analyze a set of districts or any set of geographic features (county boundaries, city blocks, etc.). When google maps shows you a city boundary, it is most likely reading from some type of shapefile to do that.
Princeton has a program ready to approximate the boundaries from a pdf or other image if shapefiles are not available, but it is a lot better (quicker and more accurate) if they can upload the newly-drawn districts as shapefiles. If the official map drawers are using any of the major software packages to draw the maps (Maptitude, ESRI, etc.), rather than sharpies, then that program automatically produces shapefiles as part of the output, so it's just a question of whether the powers that be will fork them over. Shapefiles were actually invented by ESRI, by the way, back in the 1970s, I think.