Contacts is the least well supported Google service on the iPad, at least as far as I can tell.
There is no way to edit the Contacts in the Gmail app, unless you've found something I haven't. The most you can do is add a new one when you get an email from someone new, and maybe edit that.
Adding your Gmail account to Settings > Mail, Contacts, Calendars; Add Account, will let you sync and edit your Google contacts on the iPad, however it does not support groups/lables.
You can edit your Gmail contacts somewhat by visiting your Gmail account in Safari, and choosing the desktop version of the site (usually found at the bottom of the screen, possibly in the email list. I think this works well now. Before you had to use the HTML version of the site to access and edit contacts. The free Chrome browser might work better for this, since it comes from Google; but there is no guarantee.
I am not aware of any stand alone app that does Gmail contacts, let alone does it it well. If you are anyone else discovers such an app it would be welcome.
Note: If you've enabled two-step authentication on Google, you must use the app specific password setup in Google's privacy settings to create a special password when adding Gmail to the iPad's Mail, Contacts, Calendars settings.
