1: "Scooted" has a general usage of "hurried" -- see http://dictionary.reference.com/browse/scoot?s=t -- so if you are making a dash for the toilet, you can scoot to it, sure.
2: "My eyes are weary with sleep" doesn't quite work; being "weary" means being tired, and being "tired with sleep" only works if you're delving into metaphor. (E.g., if you have a condition that makes you sleep too much, and you are always groggy and wanting to go back to sleep.) "My eyes are drooping with sleep" miiiight work? But first, it's generally your eyelids that droop[footnote1], and second, it or "heavy with sleep" are going to indicate that you were just woken up and you're still tired. If you mean that your eyelids are drooping because you are tired, you want to use "My eyelids are heavy/drooping with fatigue."
If you have eye-strain, though, you can say that your eyes are weary, or weary from [staring at the computer/reading all night/squinting all day].
[Footnote1: Unless your eyes are on stalks, in which case, please bring over your flying saucer so I can take advantage of your advanced technology to fix a few things. Thanks. ;) ]