An update for anyone in the future reading this thread.

1. The call center can in fact see nonstop flights that Aeroplan.com doesn't display. Calling in and booking was able to get me 2 confirmed and ticketed seats for the TPAC segment. They were not able to see the relevant NRT-BKK space, despite it showing on EF as well.
2. I never got an answer to this, as Aeroplan was having difficulty seeing SK/LX/NH space at the 355 mark. For what it's worth, ExpertFlyer was able to see the space each night around 8:15pm EST on day 356.
3. I did ultimately end up ticketing the one TPAC segment, and I'll ultimately add the segment to BKK 337 days out when TG space opens up. It's currently showing on Aeroplan.com, and should be a steal for an extra 5,000 miles plus the change fee.