IOS – Apple Pencil app without any analytics or syncing or network use

I use Notational Velocity and Simplenote apps on iPhone and iPad and the Simplenote web site as well and while I see you want to stick with Apple's notes.app I think you'll find Simplenote excellent, maybe better. I've been using it as long as it's been out and it's stable, simple, and has always worked for me.

If Apple mated the Notes app with something on the Mac I might never have tried Simplenote but it's cumbersome as you say to have to use Mail to deal with notes. Notational Velocity's integration with Simplenote is probably what you're looking for.

Simplenote is free, give it a go.

I've seen posts that say if you are synced to Exchange, the built-in Reminders app will sync your tasks; there is a slider for it in the Exchange setup. As far as a third-party app, try IMExchange2, it appears to sync tasks and notes.