I have an apk that requires no permissions whatsoever. Given that the apk requires no permissions, can I be completely sure that it is safe to install? Or are there still ways in which this apk might be harmful or malicious to my device, even though it does not have any permissions? Please note that this apk does not come from an official source.
In an ideal works an Android app that has no permissions would mean that the app can not seriously harm your device.
Unluckily we are not living in an ideal world, especially on Android where the device manufacturer often do provide the newest security updates to each and every device.
Therefore running an malicious app without any permission on an Android device can end up in a permanent device infection if the device is vulnerable to an exploit that allows the app to get root permissions. Those exploits can work independently if your device is rooted or not, then work even on stock rom devices.
Therefore the risk executing an app without any permission depends at large parts If your device has the very latest security update (see patch level date in Settings -> System -> About -> Security patch level).
If your device has the most recent patch level and the app is not brand new the risk is low that the app is able to harm your device. However if your device has a security patch level months or even years ago even an app without permissions may cause serious harm to your device.