Can a package uninstall itself? Can a package uninstall another package if they share the same userId and signature?
A 3rd party app cannot install or uninstall any other packages programmatically, that would be a security risk for Android. However a 3rd party app can ask the Android OS to install or uninstall a package using intents, this question should provide more complete information:
install / uninstall APKs programmatically (PackageManager vs Intents)