How do I predict a class for each time step using the information from previous timsteps I have a classification problem but  different than usual. I have to provide 3 outputs (each of them either 0 or 1) for every input of 3 timesteps and 10 features.
What model architecture or approach is used in this type of problem?

Do I predict for each time step by separating the observations?
Do I predict for each time step and for next time step also use the previous time step?
Do I set output units to be 3 and train model to predict three values for 3 timesteps?
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.