From a usability POV I definitely agree that there should be a remember me function within the logged in section of the web site. It seems logical that a user would decide they want remembering after they have seen a site is worthwhile rather than before they have ever logged in for the first time.
From a security POV this idea falls down though; too easy for someone to forget to log out and then remember me to be clicked. And to include password confirmation and all that....that would just seem a bit daft.
As to the headline question of putting the tick box before the inputs- why would this help your situation of forgetting to click login?
Surely the best place for the tick box so you remember to press it before pressing the login button is right next to the log in button?
It seems a pretty standard cognitive flow: username>password>there's the login but...what's that? remember me?....ahh....go on then>press login button.
Placing it below the login button rather than next to it...Partially for visual appearance and showing relations, having it next to the login button would disconnect the password/username from the login button with a weird tick box in between.
It also helps to minimise mouse movement- very simple to tick a box then move your mouse a few mm upwards to press login.