Yes, every subgroup has cosets. Will come back to this later.
In the second paragraph you say " as the elements not in H at least for one subgroup, that group having to be a coset". It sounds like you think that some elements outside of $H$ can form a subgroup. This is never true. A subgroup must by definition contain the identity $e$, so any two subgroups have at least one element in common, $e$. Thus it's impossible for a subgroup to be totally from outside another subgroup.
A coset is a subset of the group of the form $gH$, where $g$ is any element in $G$. Thus $eH = H$ is a coset, so there is at least one coset. If you another element $g$ outside of $H$, $gH$ is a coset as well.