Is it possible to flash an app via custom recovery? If it is, will it become a system app?
Yes, it is definitely possible and has been done many times. The method to do so is explained in this article.
Briefly, the main steps from the provided link are following:

Download any similar zip template
Extract it; there will be 2 folders - META-INF &amp; system
Make a folder with the name app inside the system folder
Paste your APK files in the app folder
Zip META-INF &amp; system into a file
Flash it