Thanks, a.lorge.
For the record, no issues with print connectivity or profiling with the HP Utility, just APS. Out of desperation, I moved the computer closer last night and connected via USB and found that APS was able to see the printer and build a profile, so it seems like it is just an issue with ethernet connection. It's a PITA workaround, but at least I don't need to profile new papers very often.
Regarding the monitor profiling, though, there isn't a different connection that is going to solve anything. It just seems to be some sort of corruption that creeps in midway into the process and forces APS to generate bad color patches, which in turn lead to horribly skewed profiles. I can see when the problem occurs, which is not always at exactly the same step in the process, but it always affects both the APS interface and the colors it produces for the sensor puck. After trying to get tech support for APS much earlier in the game, not long after I purchased my printer, I'm not going to waste my time trying to call HP about it. Especially, now that I will need to pay for the call(s). They have a "Color Team" that is the only one trained in APS and you have to beg and plead to get them to call you back. The normal first and second level techs say that they aren't even allowed to patch callers directly into those people. Once they finally call back, if they even do, they only seem vaguely familiar with it, as it is really an X-Rite product. I've had both good and bad experiences with HP tech support over the last couple of years, but the APS side of things was universally bad.