I´m wondering, if I take "Time Machine" of my Macbook Pro, does it backup Lightroom? And if it does, will my edits follow the picture?
I am not Mac user, but as long as Time Machine will backup Lightroom catalog you'll have backup of your edits.
Lightroom catalog (lrcat file extension) is a database file where all edits and metadata are stored. It is universal file, so If you move it to other machine (preserving paths to photos) you will also be able to have you edits elsewhere.