SUCCESS!!
Or at least, I've got lights blinking on the MagJack now!! This is what I did: With a sharp clean soldering tip, a magnifying glass, and steady hands, I touched each and every pin on both chips in an attempt to reflow each one in case there was a cold joint somewhere. Then, I took some desoldering braid, laid it across each side in turn, and placed my soldering tip along it, trying to soak up any extra solder. I didn't have any flux around at the time, so I didn't use that. I actually did this procedure twice. Once last night, with no success, and then again this morning. Second time's a charm! That ethernet chip feels a bit cooler now as well, since I can place my finger on it and leave it there without feeling discomfort.
As soon as I applied power, I knew something was different, since that green LED glowed brighter than it ever did before. Then, as soon as I plugged it in to my switch, voila! The green light on the MagJack is solid on, and the yellow one is blinking with activity.
Now, on to testing it through Vixen on a Smart String hub.
Thank you RJ for having faith that this was simply a bridged connection somewhere, even though I couldn't see it.