Universal family of hash functions
For your first question, here is a hint:
$$
\Pr[h(x_1) = i_1 \land \cdots \land h(x_{k-1}) = i_{k-1}] = \\
\sum_{i_k} \Pr[h(x_1) = i_1 \land \cdots \land h(x_{k-1}) = i_{k-1} \land h(x_k) = i_k].
$$

Regarding the connection between universal and $k$-universal: Consider the family consisting of all constant functions. This family is 1-universal but not universal (for $|U|&gt;1$).

However, it turns out that every 2-universal family is universal (exercise). The converse isn't true: for example, the family of all functions $h$ such that $h(x_1) = 0$ is universal but not 2-universal (for $m&gt;1$).