Google can delete and install apps silently. If someone hacks this mechanism, an attacker can install arbitrary apps on a device. Unsure if this app has automatically all rights it wants to have. This doesn't happens till now, but it's possible. You can only protect your phone by checking regularly all installed apps and there rights.
Delete may be silent, but the only push-install mechanism I know about is to use the firmware update mechanism, and that is not silent - the phone reboots twice.
Malicious applications can do a lot evil things, but if you don't give every app the rights the apps wanted and think a little bit, you can protect your phone.
Unfortunately, you cannot decline to grant permissions, you can only decline to install an app that requests more than you want to grant. And some permissions are quite broad, so an app may legitimately need a tiny part of some permission, but get the whole thing.
Some security holes in the browser or the system allows an app to get root-access. In this case the app can do everything it wanted. I don't know any protection against this. As far as I know android has such security holes, so this is the most dangerous issue.
On the other hand, these security holes are unfortunately what the owners of most devices must rely on in if they wish to to be able to install custom firmware, since most consumer phones were not designed to permit this. Knowledge about android security exploits is not much behind that of conventional linux distributions (most of the root exploit are inherited linux kernel exploits, not android exploits, since android has few components with root authority).
