Interesting point. I can't remember the last time I had to search while offline. Not to say that it'd never happen, but implementing a decent index on a mobile device doesn't sound trivial.

I assume this is their web stack. For their mobile apps they most likely use native iOS / Android codebases. The typical stack for a data heavy native app is to use an SQL store (or just filesystem) to cache data locally so its available offline. You won't obviously be able to search all of your emails offline though, only your most recent ones that happened to be loaded during online usage.

They don't, it's all online with webviews on iOS and Android. I don't think there is offline functionality (still haven't gotten an invite).