In a custom application with two custom profiles, the profile settings were added to the package. This package was installed in a developer org. In the new version of the app (package), if we add/remove settings in profile, would those be treated as push upgrades in managed package? For example: Adding a new object. Including this object's page layout and record type settings in custom profile.
You'll want to read About Permission Sets and Profile Settings. For new permissions:
Profile settings are applied to existing profiles in the subscriber's organization on install or upgrade. Only permissions related to new components created as part of the install or upgrade are applied.
So, if you add a new component, and update the profile at the same time, they will be updated. Otherwise, they will not be updated, because the subscriber could have modified the profile in the interim.