How can I sign in under my google account for the mobile version?
I signed in via my google account on my laptop. When I search for http://todoist.com/m it brings up a sign in page. Can I sign in under my google account here, too, or do I have to make a todoist account to use the mobile version?
you don't need to create a new Todoist account, but you'll have to set a new password only for the mobile access. To do so, log into your account through the web-app, go to Preferences (gear icon in upper-right corner) -> Account and use the "Edit" link to set a password which you'll be able to use in combination with your GMail address on the mobile site.
Note - you'll still be able to sign in with 1 click to the web-app using your Google Account, but the mobile site will require the use of the new password.