Reinstall iOS preserving apps and data
Yes. By doing an iCloud backup (not iTuens) you only preserve apps, data and most basic settings. Phone will look the same after you recover from an iCloud backup.

Doing the same but with local Itunes backup will preserve absolutely Everything.

To reinstall ios on the iPhone you can either connect it to Itunes and do the procedure from there or directly on phone by going:

Settings --> Generall --> Reset --> Erase all content and settings.

Unfortunately this is not documented anywhere so i can not reference it but from personal experience this is my knowledge.