How can I find a compatible device for a given app on Play Store? There was an app (Carrot Fantasy, already removed) that I can't install directly from the Play Store on Android tablets. It does run on a Samsung Galaxy Note 3 smartphone, but I have not yet had any luck with any tablet I have tried.
I managed to have the Play Store on a Google Nexus 7 shown the app, but it would not offer an Install option and just said This app is not compatible with your device. without any further information.
On other devices such as Samsung Galaxy Note 10.1 2014, the app does not even show up in Play Store.
I have searched for other similar questions, but they usually seem to be asked from the developer point of view, i.e. how to modify the app manifest or how to check one's settings in Play Store. They did not provide any insights into what to do on the user side.
Certainly, there must be a way for me as a user to diagnose why the app is deemed incompatible, what specific requirements it has, and then pick an Android device that fulfills those requirements - how?
It appears that you have already tested the surest way.  Attempt to download the app and it will specify its compatibility.  You still might not be absolutely sure until you actually try to run the application.
As mentioned in the comment, you can ask the developer or communicate with others who have used the app (via feedback on Google Play or the author's provided web site).
The only way the author can be totally sure is by testing it on every Android or getting feedback from their users.  There are many occasions where based on feedback, the author has to go back to the drawing board to work on glitches to support some devices originally thought to be compatible.
I have a number of Androids.  I just clicked on the install option and find that it's compatible with my Samsung Galaxy S3, my very old phone (the Samsung Galaxy Prevail).  However, it's not compatible with my Samsung Galaxy Tab7 or my Acer-7 tablet.
By the way, good question.  It really appears there should be something specific since Play Google is using some criteria to display compatibility.