Since I'll be vacationing in Japan for a week where there's no BIS service, despite I may get on webmail and cleanup my emails, when I return home, wouldn't all the emails since the last time I shut down appear and I'll have to delete them all again?

I'm not sure I'm fully understanding your question....

But if you're logging into the web client and deleting your messages than why would you still get them when you returned home? They'd be gone.

They would, but how much hardship is to use Delete Prior and wipe all emails by a single push of a trackwheel?