Starting from March 2019, is it necessary to build all existing apps using iOS 12 SDK and Xcode 10 for it to work on iOS 12 devices?
The key words in this announcement are "new apps and app updates".

Apps which are already available on the App Store do not need to be rebuilt with the new SDK. Apps which were built with earlier SDKs will remain available on the App Store, and will still be usable on devices. However, after March 2019, the developers of those apps will need to rebuild their apps with the iOS 12 SDK to release any further updates.