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What happens if I log out of Google, clear all cookies/cache (or even use a new computer to login to Google)? Will Google allow my login with no previous login data stored on my computer?
You would likely be treated as a first time user of any Google site you visited until you had a cookie placed on your computer. The website experience for a first time user would be different than for someone who has used their site before, such as a new user might be required to consent to TOS or cookie usage. If you have used the site before, Google will place a cookie on your browser so these messages or warnings do not appear.
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You need to type your user id and password to login again if you have cleared your cookies...if you don't clear your browser cookies then you have to press just "sign in" button.
I'm sure they would have a way to track your IP when you login again and are not simply relying on temporary cookies which can be so easily removed from a computer.
Yes, if you once clear your cookie, then you have to login again.
You will only need to login again. This is the situation with most sites. Though I understand what you're asking. If you have not logged into your email account for a very long time, then you might have to confirm via mobile or alternate email address.