can the corp choose to trash an asset in order to install a facedown card for a given remote server? I was under the impression that assets remained until the runner trashed them or they had a specific ability that caused them to be trashed...

Yes. When the Corp installs ICE in front of a server, they may trash any ICE already in that server. When the Corp installs a card into a server, they may trash any cards already in that server. This means you can trash an Asset to install an Agenda.