AlphaGo Zero loss function
The best way to understand that part is by looking at figure 1 in the AlphaGo Zero paper.

The neural network (NN) minimizes the differences between its own policy $p_t$ and the MCTS policy $\pi_t$. The value of $\pi_t$ is produced by the MCTS self-play which in return uses the NN from the previous iteration.

The same goes for $v_t$ and $z$. In each iteration the weights of the NN are adjusted to minimize the distance between $v_t$ (output of the NN) and $z$ (output of the MCTS) as defined by the loss function. $z$ does not have a time index here as the full self-play produces just a single value for $z$ each time it is conducted.      

TLDR for your first question: Both, $\pi$ and $v$, are being produced by the MCTS as input to the NN. 

(The indexing in the paper is a bit confusing in my opinion so it is probably easiest to just look at it as stated above)

Now, with "input" I do not mean input on the input layer of the NN. As described in the appendix under "Neural network architecture" the input is a "19 x 19 x 17 image stack". which contains the following information:


The positions of player 1 for the latest 8 rounds (8 feature planes)
The positions of player 2 for the latest 8 rounds (8 feature planes)
A color feature indicating whose turn it is (1 feature planes)


And those 17 feature planes ($8+8+1$) combined with the $19\cdot19$ sized board is the $19\cdot19\cdot17$ input the NN receives thru its input layer. $\pi$ and $z$ are passed to the NN via the loss function only (i.e. they are the target values in this supervised learning problem!).

TLDR for your second question: $\pi$ and $z$ are not fed to the NN thru the input layer but just via the loss function as target values.