if you are logged in (Google Account) it is kept permanently in your account for more relevant search results based on your Web History. (even if you change your IP number, if you are logged in, search results will be based on your previous history)
if you don't have a google account (or you are not logged in), you information will be kept in a small text file (stored in your own computer) we call "cookie".
If you want to search and browse the web privately, you can activate the "incognito" mode in Google chrome or "in-private" mode in internet explorer. you can check this article for more information: https://support.google.com/websearch/answer/4540094
if you are not logged into a Google account, information is stored in your own computer, in a text file called "cookie" (cookies)
yes, from this page www.google.com/history you can Access all of your past searches.
no, it is not stored anywhere else. it is only stored in your own computer. if you want to delete your web history from your computer you need to go to settings > history and then click the "clear browsing data" button one time.
it looks like this
from another computer, if you checked the "Stay signed in" checkbox, YES, it will appear on the list of options of accounts to sign into. If you never checked this option, your account will not appear.
(can you select "i don't know my username" and continue)