Myhill-Nerode classes for a Language $L_k = \{x1y | x \in \{0, 1\}^*, y \in \{0, 1\}^{k-1}\}$
First, I think you have to review the Myhill Nerode Theorem and its relationship to the states of finite automata. You seem to think the number of equivalence classes is the same as the number of states in the minimal NFA, but it's actually the same as the number of states in the minimal DFA.
As a side note, minimization of NFAs is apparently much trickier than minimization of DFAs, so in an introductory formal languages class, you're not going to hear much about minimal NFAs (except in cases like this, where it's straight forward).
So this means it's parts (a) and (b) that are closely related.
As this is a worksheet question, I'll try to only give a guiding hint, rather than a straight-out answer.
The key thing (at least in my opinion) is to realise that each language $L_{k}$ is that it is only really concerned with the last $k$ symbols in the string. So the number of equivalence classes is closely related to this, and the proof that the obvious guess here works is informed by this observation.
This same observation is what gives the minimal DFA (unsurprisingly, as the equivalence classes exactly correspond to the states!). As the DFA doesn't know when the end of the string is coming up, one way you can think of it is as a little sliding window algorithm, and you should be able to relate the states, transitions and equivalence classes in terms of moving this window along one symbol.
To get the minimal NFA, lean on the nondeterminism. What is the piece of information that the DFA doesn't have that you can just nondeterministically guess?