Will changing an iCloud profile on iOS harm data on the previous account? I share an iCloud with my mom. If I delete iCloud on my phone, and created a individual one, Will it delete her data?
Delete iCloud? You mean sign out I suppose. If you want a separate account without harm of data, Go to Preferences > iCloud and somewhere there are sync options. 

Disable synchronization (possibly other features), sign out of iCloud, then create a new profile and use it on your iPhone. You might need the previous password to continue.