Do you not need the Google account? It sounds like you just don't know how to adjust your settings properly. What, exactly, are you trying to turn off?
If you want to turn off mail for that account then go to Settings->Mail, Contacts, Calendars, select the account, and turn Mail off.
If you want to receive mail but don't want the notifications/alerts then go to sounds and set Mail to None and/or go to Notifications and adjust the notification settings for Mail.
If you want to pull instead of having gmail pushed then make sure you're adjusting settings at both Settings->Mail, Contacts, Calendars->Fetch New Data and Settings->Settings->Mail, Contacts, Calendars->Fetch New Data->Advanced->(your gmail account).
You don't have to remove the account if you want the contacts synced from the account and syncing with the account (or another account with the contacts you want transferred to that account) is really the best way to have the contacts on your device. Otherwise if you lose your device you lose your contacts stored only on the device.
Nope. Contacts synced from Google are only synced with Google. They don't sync with iCloud as well. If you remove the Google account it removes the Google contacts. For what you're describing the contacts would have to be migrated to iCloud.
The point is always subjective. My wife used Google contacts but didn't use gmail for years. Don't just assume that your preferences and situation are universal no matter what the topic.
