Confused on Variational AutoEncoder
VAE's have an encoder.  The encoder maps an input to a latent vector. The latent vector doesn't "stick".  Instead, in each iteration, we use the encoder to map to a latent distribution, sample from that latent distribution, then use the decoder to map to a reconstruction, and compute the loss.  Yes, each data point is mapped to its own latent vector.  Yes, you run the encoder anew and sample anew for each point and each iteration of training.

You check for convergence in the standard way for any neural network: you look for the value of the loss function to plateau.