How to keep useful but rarely used apps uninstalled in the phone (ready-to-install, ideally including their settings)?
You can go to Settings > Backup &amp; reset > set Back up my data to ON. Set/select a backup account, then enable Automatic restore.

If the phone was rooted, Titanium Backup is probably what you should try.


  You can backup, restore, freeze (with Pro) your apps + data + Market links. This includes all protected apps &amp; system apps, plus external data on your SD card. 


From its website:


  Titanium Backup is a backup utility for Android that backs up your system and user applications along with their data on external storage of your choice


Edit: OP could also use adb backup as described on this answer

Edit 2: Below is the response I received from Google Product Forum after asking if the backup data of a removed app are removed during the next sync,