Which modelling method to use for this time series data?

I have a time series data of reaction times (RTs) from a psychophysics study. The RTs are influenced by whether a stimulus is presented to the left or right (binary predictor). Although the RTs are mostly driven by the current presentation (at time t), I want to know how much previous presentations (t-1,t-2,...) affect the current RT. The RTs also generally become faster over time due to the subjects becoming better at the task.

I was considering using a some sort of a shifting window multiple regression. I would set a certain window size (let's say 5), and then I would use the presentations at t,t-1,t-2,t-3,... as individual binary predictors to predict the current RT. I would find the coefficient for each predictor by shifting the window by a single trial and continuing until the end of the time series. I will also need to have a predictor that accounts for the tendency for RTs get faster over time.
I have some concerns over this method:
  1. I have multiple subjects who did the task. Would I calculate the regression coefficients for each subject and average them? Or would it be better to pool all the data together and construct a single model?
  2. The predictors will all be binary except the time predictor. Is this okay for multiple regression? Are there better alternatives? (I was considering MANOVA or Markov model)