Rake before Mangle

If you apply Rake before Mangle, does the bleed buff still affect Rake?
Yes. You still want to mangle before rake if you don't have any similar buffs that'll be up instantly, because the initial hit also counts as a bleed and is increased as well.
Like Dagglet said, the initial Rake hit counts as a bleed - so that's why people recommend you Mangle first, since it increases that initial hit's damage.