Are the elements of a set within a set also the elements of the latter? It is my understanding that an event is a subset of the set of all possible outcomes (sample space). If however the sample space consists of elements which are sets, can an event be defined as one the elements from these "inner" sets?
Ex. A coin is flipped twice, (S={(H,T),(T,H),(T,T),(H,H)} Is the event A=(H) valid for the sample space despite not being a subset of S?

No.
In the first place, if $x\in y$ and $y\in z,$ then in most cases it is not true that $x\in z.$
For example, consider the set $\{\  \{1,2,3\},\  \{2,3,4\}\  \}.$ This set has only two members. If $1,2,3,4$ were members of it, then it would have at least four members.
In the second place, in the set $\{\,(H,T),(T,H),(T,T),(H,H)\,\},$ the pair $(H,T)$ is not a set with members $H$ and $T$; rather it is an ordered pair with components $H$ and $T.$
