Weather-ProofingI found the missing rubber strip for the driver's door, and it's actually in pretty good shape, so that's one less $30 tidbit to buy. And I was able to press the passenger's side piece back into place. The doors are fine now.
The brand new soft top in the trunk looks great, but the installation process is much more involved than I realized. I'm missing some of the necessary fasteners, too. I think, in the short term, my best bet is to keep the hard top on it and to rig a plywood & sheet plastic "plug" for the rear window space until I can get the glass replaced.
About the rear window: after checking with a Triumph forum for ideas, I had a couple options. I found a place that sells replacement glass for $300+, but I also found Lexan sheeet at the local Home Depot for about $80. And then, I checked with my local glass shop. They can fabricate a flat automotive laminate glass window for only about $90. That's clearly the best option. I need the rubber window seal ($120) before they can make the piece, and a little searching turned one up at Spitbits.com. While I was there, I ordered the missing fasteners ($3.60) for the new top.
Battery & StarterThe battery is truly shot, not taking a charge at all. I'll pull it before heading to work today, so I can get the core discount on the new one ($79 with core) at NAPA.
With the charger pumping out the 100A "start" current, I tried the starter. It spun freely but wasn't engaging the engine. According to my Haynes manual, it's either loose on its mounting, or the pinion is sticking on the sleeve, or it's shot. I'll pull the starter and inspect it before I just buy a new one ($50 at AutoZone).