My neighbor, who is a retired diesel/heavy engine mechanic (used to work for Cat) and who has his own little stable of heavy equipment (grader, dump truck, forklift, backhoe, and more) came over this evening.
1) The dipstick showed nearly no oil. We put 5 quarts in to get it to a healthy range. Apparently when that oil cap blew off, I lost a lot more oil than I realized. Uh-oh. Still not sure why the oil cap blew off in the first place. There may have been an underlying or pre-existing issue that caused that.
2) He had me run it, and did some listening and feeling along the engine block. His diagnosis is that it most likely has a messed up rod or rod bearing inside the engine, causing one piston to not compress fully, thus generating the white exhaust and ugly noise.
Apparently "take apart the engine for repair" is such a big chore that it may be easier/cheaper/safer to drop a replacement engine in there. He is not sure he wants to do the job.
I personally am not equipped to do it (knowledge/time/engine lift). He says he'll help, but I'm not sure how long it'd take. This may be something I'm better off outsourcing to the local dealer.