Is it possible for the corp to willingly trash Architect? Specifically, if Architect is installed on a server, can the corp choose to trash it when installing a new piece of ICE in that server?

The rules say:

"When installing ice, the Corporation can first trash any ice protecting that server in order to reduce the install cost."

And Architect says:

"Architect cannot be trashed while installed."

That reads to me like once it's installed, it can only be removed by something like Blue Sun's ability. Is that correct?

Correct. Prohibitive rules always take precedence over permissive ones. Architect may be uninstalled, but never trashed.

EDIT: In the interests of completeness, an UNREZZED Architect may still be trashed however, as its ability text is not active.