How do you know if your Autoencoder network is fully connected?
Is Autoencoder with more than one hidden layers a type of deep neural
  network (DNN)?


Yes.


  Is DNN always fully connected?


No. For example Echo State Networks are sparsely connected DNNs.


  how do you know if the Autoencoder network that you build is fully
  connected?


I am not sure about R because I don't use it but for Keras/Python you can read the layer descriptions. The most common predefined layers (Dense, Convolutional, RNN, LSTM..) are fully connected. If you want to have a sparsely connected layer, then you can probably create a custom one using a Lambda Layer.