Ok, just to plug in my own experience:

1. Bottom line is that all went well. Made the changes, while on the phone confirmed that they went through on the UA site, also received emails confirming them. To be triply sure, I also checked the SQ and TG sites, and called both of those other airlines to confirm that they show the itinerary as ticketed (since I'd read occasional reports of UA making award reservations involving partner airlines but messing up the ticketing).
2. CSR seemed to know what she was doing. But when I first asked about the TG flights I wanted, she went ahead and started a new reservation just involving those flights. When I then said I wanted to change my existing reservation, she politely cautioned me that this could have caused problems in terms of her having grabbed the award seats and their not becoming immediately available again for the reservation change. It turned out not to be a problem, but what I think I learned is that under these circumstances in the future I'll tell the CSR just to check availability, not to start a new reservation.
3. The one aspect of the customer service that fell short was that she put me on hold for about 30 minutes without checking in with me to let me know what was going on. The wait I half-expected; her not checking in was a bit disappointing. I'm unclear on whether all this time was involved with getting the TG flights or her confusion (which she mentioned to me) about what an R reservation code on SQ meant.
4. FWIW, she was based in Tampa.

Thanks again for everyone's help! Now, back to checking these reservations weekly to make sure they stay put!