new language accepted by FA when new transitions are added to a FA?
Your answers to (a), (b), and (c) seem correct to me. Here's something to think about for (d).
Hint for (d): Let's say $s = s_1s_2...s_k \in L$, and on input $s$ the original transition function follows the path $P = q_0, q_1... q_f$. Let $q_{s_i}$ denote the state $q_i$ closest to $q_f$ in $P$ after only reading up to $s_i$ in the original NFA. What are the strings can I generate by doing the following in the NFA defined by (d)? $\lambda$-transition from $q_0$ to $q_{s_i}$, follow $P$ up to $q_{s_j}$ (for any $1 &lt; i &lt; j &lt; k$), and then $\lambda$-transition to $q_f$ and accept.