IOS – see how many times I reinstalled an app, after I delete it

Yes, you're just syncing the App's data with iCloud, as no files are leftover when an App is deleted.

iOS apps (and now OS X apps too), are Sandboxed. This means that each App has its own space in disk, with its own directories, which act as the home for the App and its data.Deleting an app from the iPhone deletes this sandbox, deleting all data associated with the App.

So, if an App supports iCloud sync, whenever you install that App with your Apple ID, in any of your devices, all the information will be synced.

I found how to make iTunes restore only contacts, while preserving the rest intact:

1. run iTunes before connecting iPhone
2. disable auto-syncing in preferences: Edit > Preferences > Devices > Prevent iPods, iPhones, and iPads from syncing automatically should be switched On
3. connect iPhone to PC. In iTunes:
4. on the Sidebar, select your iPhone (press Ctrl-S if Sidebar is hidden)
5. on the top central menu (Summary | Info | Apps | Music | ...), click Info
6. scroll the page to bottom, there's Advanced section
7. switch on: Replace information on this phone > Contacts
8. click Apply in the bottom right of iTunes window (or right click on iPhone and choose Sync)

These steps wipe the local contact store on iPhone (leaving cloud contact sync alone, if applicable) and injects all the contacts stored in the PC-side data source that iTunes has selected (from Windows Contacts in my case; Outlook on Windows / Contacts on Mac in other setups) and pushes those contacts to the device. It does not load contacts from an iOS backup.