Can a security app check what servers(urls) is another app is trying to connect?
I know about android sandboxing but is there still a way using PackageManager or any other API/technique to get this piece of info programmatically using public SDK only, for Non Rooted android phones?
As far as trying to know what other app is trying to connect to is considered security / privacy violation, if that is allowed you may as well dump individual request.
So it should not be possible to achieve that on a non-rooted android phone.
If you are looking for a way to achieve a similar functionality set the android device to use a proxy / gateway controlled by you, and filter / track URL requests there.
If you will root the device then there are many other possibilities.