Ah, I see what you mean. Yes, we do have a “done” setting for notes, but it indeed can’t be searched on yet, so is not that useful. We do have plans to add that search option later.
To track notes in the future, you have a few options: you could add a tag to these notes in the text (eg #future). Then you could Search All for #future and save it as a saved search “Future” (premium feature).
Other ways to deal with it might be pinning the note to the top of the project, or setting a future date so it appears at the top.
You could also make a saved search for a range of dates, such as the coming 7 days, and then give your notes dates. When they are coming up, they will appear in that saved search.
The iOS app also has saved searches, yes.
The searches you can make are the same as in the free version. You can search on date range; on-the-agenda setting; text; and tags/people. Basically, any search you can make in the free version can be saved as a automatically updating overview.
See Searching and Filtering