If you are being asked for your current password, you arent doing a password reset. You are clicking to change your password. If you are unsure what your password is, you need to log out, and then where you log in there is a forgotten password link
Check you have it enabled for login on the front end.
In terms of the last question, they will not appear like that on our site. Thats on googles site. When they click to login with google however, they will see googles site with their logins
There is nothing within the platform at present that can give you a list as such. If you suspect someone is using multiple accounts you can use the IP Tools section in the moderator CP or admin CP to see if there are multiple accounts
I would advise on contacting your hosting provider for assistance with this. The index file is the root to all pages on your site, so that wouldnt nessesarily be an indication its not mysql.