Independent with a vector v.s. independent with its components Suppose that $Z$ (a scalar) is independent with $X=(X_1,\ldots,X_n)$, $n>1$. Then, $Z$ is independent with $X_1,\ldots,X_n$ because each of the latter is a function of $X$.
I suspect the reverse direction: $Z$ being independent with $X_1,\ldots,X_n$ implying $Z$ being independent with $X$ is not true. Most likely because we would need some info about the joint distribution of $X_1,\ldots,X_n$. But I can't think of a counter example. So could you please provide one as well as some intuition on how you arrive at it?

I is not only about the joint distribution of $(X_1, \dots, X_n)$.
It is possible that all components of $X$, namely  $(X_1, \dots, X_n)$ are independent and that all pairs $(Z,X_i)$ are independent, but $Z$ is not independent from $X$. 
My favorite example to illustrate this is from Bauer's book Wahrscheinlichkeitstheorie (it's a german book):
Suppose you throw a fair dice two times and let $Y_i$ denote the outcome of the $i$-th throw. We define:
$$
X_1= \begin{cases}1, & \text{if} \ Y_1 \ \text{is odd} \\ 0, & \text{if} \ Y_1 \ \text{is even} \end{cases}, \quad
X_2= \begin{cases}1, & \text{if} \ Y_2 \ \text{is odd} \\ 0, & \text{if} \ Y_2 \ \text{is even} \end{cases}, \quad 
Z= \begin{cases}1, & \text{if} \ Y_1+Y_2 \ \text{is odd} \\ 0, & \text{if} \ Y_1+Y_2 \ \text{is even} \end{cases}, 
$$ 
Clearly $X_1$ and $X_2$ are indpendent and it is also easily verified that $Z$ is independent from $X_1$ and that $Z$ is independent from $X_2$. However $Z$ is not independent from $X$.
This easily generalizes to $n \geq 2$: you throw a fair dice $n$ times and define $X_i$ as above and $Z$ is again defined via the sum of $Y_1, \dots, Y_n$.
The intuition is that knowing the results of all $X_i$ except one, say $X_n$, the outcome of $Z$ and the outcome of $X_n$ contain the same information.
