Time series prediction
Let me give you a few simple approaches in time series analysis.

The first approach consists in using previous values of your time series $Y_{t}$ as in $Y_{t} = \phi_{1}Y_{t-1} + ... + \phi_{n}Y_{t-n}$. In case you don't know, these models are called autoregressive (AR) models. This answers your first question. Of course it is useful to include the previous value of your time series. There is a whole set of models based on that idea.

The second approach is taking a window and extracting some features to describe the time series at each point in time. Then you use a conventional machine learning technique to predict future values as typically done. This is more common in a classification or regression setting but future values can be thought of as classifying future values. This technique has the advantage of dramatically reducing the number of features, although you usually lose characteristics associated with time. This addresses your second concern.

Another model that could be helpful in your case is the vector autoregressive model (VAR) (using Wikipedia's notation):

$$\left( \begin{array}{ccc}
y_{1,t} \\
y_{2,t} 
\end{array}\right) = \left( \begin{array}{ccc}c_{1} \\ c_{2}\end{array}\right) + \left( \begin{array}{ccc}A_{1,1} &amp; A_{1,2} \\ A_{2,1} &amp; A_{2,2}\end{array}\right)\left( \begin{array}{ccc}
y_{1,t-1} \\
y_{2,t-1} 
\end{array}\right) + \left( \begin{array}{ccc}
e_{1,t} \\
e_{2,t} 
\end{array}\right)$$

Here you can see that $y_{1,t}$ has a contribution from its previous value $t_{1,t-1}$ but also includes the value of the other series $y_{2,t-1}$ in a linear combination. As usual, the purpose is to find the elements of $A_{i,j}$ that minimize some measure of error between observed values and estimated values.

A general suggestion: The first thing you need to do is to test the autocorrelation of your first series in order to confirm that an autoregressive approach is suitable and then test the cross correlation between both series to support the idea that using the second series to improve your predictions is appropriate.