How do I predict a class for each time step using the information from previous timsteps
Regarding neural networks, and supposing your output is based on three prior timesteps, two possibilities could be:

An approach with a CNN model with a (3,10,1) input and your desired output.
LSTM where you use three cells the 10 features as an input for each timestep and output the classes you need at the end (don't know if this one will work properly though,maybe it is too small, depending on your problem).
As in you question 1. you propose you could use a DNN with the input of one of your previous timesteps, but I don't think this will be the best approach neither.

If you provide more details (like an example of your dataset) we can help you with more specific answers.