How can autoencoders be used for clustering?
Clustering is difficult to do in high dimensions because the distance between most pairs of points is similar.  Using an autoencoder lets you re-represent high dimensional points in a lower-dimensional space.  It doesn't do clustering per se - but it is a useful preprocessing step for a secondary clustering step.  You would map each input vector $x_i$ to a vector $z_i$ (not a matrix...) with a smaller dimensionality, say 2 or 3.  You'd then use some other clustering algorithm on all the $z_i$ values.  

Maybe someone else can chime in on using auto-encoders for time series, because I have never done that. I would suspect that you would want one of the layers to be a 1D convolutional layer, but I am not sure.

Some people use autoencoders as a data pre-processing step for classification too.  In this case, you would first use an autoencoder to calculate the $x$-to-$z$ mapping, and then throw away the $z$-to-$\hat{x}$ part and use the $x$-to-$z$ mapping as the first layer in the MLP.