DFA for Strings with third symbol from RHS as 1
You probably want to substitute $a$ with $1$ and $b$ with $0$ but apart from that your solution is correct.

Finding an equivalent NFA is easy with Thompson, Glushkov etc. From that you can derive an DFA via powerset construction. However, for this short expression you also can just write the DFA down. Imagine the states as storage what the last 3 symbols were, and accept if the third last was a 1, i.e. name the states 000 to 111 where 000 is the initial state and all states with a leading 1 are accepting. A transition is just a shift-left with the last read symbol on the right, i.e. $\delta(abc, d) = bcd$, with $a,b,c,d \in \{0,1\}$.