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, but you can get something close.  When you're looking a series of outcomes like that, the sample space is the product space of the sample space of each individual experiment.  Here, for example, your sample space is $\{H,T\}^2$, and if $\pi_i(a_1, \ldots, a_i, \ldots, a_n) := a_i$ is the $i$-th component function, you can arbitrarily define the event $E = \pi_1^{-1}(H)$ to get the event you're looking for.
More importantly, this set is well-behaved, meaning you can always find its probability (i.e. in terms of measure theory, it's measurable with respect to the product sigma algebra).
