Primarily a todo app, but it does everything you are asking for.
It's cross platform
You can export your lists
The search is amazing and instant
You can create projects, categories, labels, and add new notes with a hashtag to these projects - and you can add files in comments to each "task", which you'd store as a title for your document
It works offline - activates offline mode automatically as soon as an internet connection is no longer detected on any platform you are on such as iOS, Android or Web.