You can do time-series prediction with neural nets, but it can get pretty tricky.
The obvious choice is a recurrent neural network (RNN). However, these can be really difficult to train, and I would not recommend RNNs if this is your first time using neural nets. Recently there has been some interesting work on easing the training of RNNs (e.g. Hessian-free optimization), but again - it's probably not for beginners ;-) Alternatively, you could try a scheme where you use a standard neural net (i.e. not a RNN), and try to predict the next frame of data from the previous? That might work.
This question is too general, there is no categorical right answer. Yes, you can use unsupervised feature learning as part of your solution (e.g. pre-training your model), but if your end goal is time-series prediction you will need to do some supervised learning too.
There has been some work on adapting deep learning methods for sequential data. A lot of this work has focused on developing "modules" which can be stacked in a way analogous to stacking restricted boltzmann machines (RBMs) or autoencoders to form a deep neural network. I'll highlight a few below:
- Conditional RBMs: Probably one of the most successful applications of deep learning for time series. Taylor develops a RBM like model that adds temporal interactions between visible units and apply it to modeling motion capture data. Essentially you end up with something like a linear dynamical system with some non-linearity added by the hidden units.
- Temporal RBMs: In his thesis (section 3) Ilya Sutskever develops several RBM like models with temporal interactions between units. He also presents some interesting results showing training recurrent neural networks with SGD can perform as well or better than more complex methods, like Martens' Hessian-free algorithm, using good initialization and a slightly modified equation for momentum.
- Recursive Autoencoders: Lastly I'll mention the work of Richard Socher on using recursive autoencoders for parsing. Although this isn't time series, it is definitely related.