How are Bayesian Nets, Hidden Markov Chains, Conditional Random Fields and Neural Nets related?
No, needing training has nothing to do with unsupervised/supervised. Supervised learners need labels, unsupervised learners do not (they do things like clustering, density estimation, dimensionality reduction). In fact you can use all of them for supervised and unsupervised learning, except maybe CRFs, since an unsupervised CRF would more likely be referred to as a markov random field (MRF) since there is no conditional per se. Someone might argue with me on this last point and I probably wouldn't argue back.
No, CRFs have undirected edges. ANNs can have undirected edges also (for example autoencodes with tied weigths). ANNs don't need have a probabilistic interpretation.
Sometimes it would be obvious. For example, if I'm doing Image Denoising I'm not going to use an HMM. Other times it will depend on things like what the practitioner has experience with, what works well in this domain, or a myriad of other factors.
Out of all the thing on your list ANNs are the only one that need not have a probabilistic interpretation. Furthermore, the techniques used for training ANNs normally are of a different flavor than what is used for other methods. That being said there are lots of classes that will focus on other types of models and likely not discuss ANNs, for example Probabilistic Graphical Models, Generalized Linear Models, etc.