Is this decidable language a subset of this undecidable language?
No, A(DFA) is not necessarily a subset of A(TM). When you write "(M, w)" you actually refer to some encoding of M in some alphabet (usually {0,1}). These encodings of DFAs and TMs can be very different. 

For a simple example: Let $A_1, A_2, \dots$ be an enumeration of all DFAs, and let $M_1, M_2, \dots$ be an enumeration of all TMs. You could say that you encode $A_i$ as the word $0 \text{bin}(i)$ and $M_i$ as $1 \text{bin}(i)$, where $\text{bin}(i)$ is the binary representation of the number $i$. Then the two langauges you presented are disjunct.

As it seems that this is part of your question: Being a subset and being computationally easier do not relate. Take the set $\Sigma^*$ of all possible words. It is easily decidable (even by a DFA) but every language, even undecidable ones, are subsets of it.