Can the app see my wi-fi SSID?
Can it get my internal and/or external IP address?
My physical coordinates?
In addition, if two or more devices are connected to the same wi-fi network, would it be possible for the app to "know" that both of those devices are connected to it?
An app can actually do all that, except from knowing your physical coordinates, without requiring any permissions. This is evident on an app I use, Net Analyzer. The only permission it asks for is Location. When revoking this permission, it can still see a large amount on information regarding your network such as internal/external IP addresses, MAC addresses, other devices on your network, the services running on these devices, etc.
Note that without any permissions your external IP address can be identified - this gives the opportunity to locate your position, although it won't be very accurate.
To stay safe from an app obtaining too much information, I would recommend disabling unnecessary or dangerous permissions as shown on this article. Some app developers show their permissions along with an explanation of why they are required which is a good way to identify why the permissions are needed, and what permissions can be denied without affecting the performance of the application. It's very difficult to know exactly what an app is doing behind the scenes, and the only real way to know is to decompile the app and view the Java source code.