For your first question please have a look at the following article as an example: https://code.tutsplus.com/tutorials/wp-rest-api-creating-updating-and-deleting-data--cms-24883. It’s just the first article Google showed for “wordpress rest create new post”. Basically, all you need to create a new post is to use WP REST API. You should consider authentication mechanism. The staring point for you is JWT Auth plugin for WP https://wordpress.org/plugins/jwt-authentication-for-wp-rest-api/

As for your question about push notifications, you may consider using OneSignal for this. So, you need to use WP hooks to trigger OneSignal API on new post creation https://documentation.onesignal.com/reference#create-notification. As well as use OneSignal plugin for Ionic to implement such notifications in your app https://documentation.onesignal.com/docs/ionic-sdk-setup.You’ll be required to create account on OneSignal and create new App inside which belongs to your WP/Ionic app.