Recommendations for note-taking app for Iphone that does NOT sync to the cloud
No need for another app, there is a solution for syncing the iOS Notes app without using iCloud or any other third-party service.  Since you mentioned WebDAV, I assume you are willing to provide your own server.  It takes a little bit of setup, but if you create your own email server (specifically an IMAP server), Notes will sync to it.  There is no need to support sending mail at all, just IMAP, with which you will be able to create folders.  Just add new email account and turn on the "Notes" option for the account.  I believe the server version of OS X will even has an email server built in with easy setup, though there are many other good options such as courier or dovecot.  If you are happy only syncing at home, you don't even have to expose the server to the public internet, or you can get a domain name (or dynamic DNS service) and sync to your home server wherever you have internet access.  Note that new features in iOS 9 notes, such as handwriting and lists, are unsupported in IMAP notes accounts, but everything doable in iOS 8 is still possible including attaching images.