I don't think that approach is appropriate with time series data because the observations are not independent and you can't simply extract some of the data into different sets or you would destroy the integrity of the data.

I agree, I believe if he is taking the moving average to find the level he should be dividing by 4 not 8, however, it appears that he made 2 steps summaion. 1- He summed 4 periods, then he sumed 2 of 4s which I didn't really understand why

QUARTILE is an old function just kept for backward compatibility reasons. See explanation here: support.microsoft.com/en-us/office/quartile-function-93cf8f62-60cd-4fdb-8a92-8451041e1a2a

Hi. 1. Yes, you apply the same idea because the parameters are with respect to the stationary time series and with some algebra you will get the forecasted value for the original non-stationary time series. 2. You first fit the AR(p) model. Calculate the residuals and use it as the proxy of your errors. Now you have 2 options, to use your series of your residuals as proxy to the errors or calculate the variance of the residuals and simulate the errors from Normal distribution with mean 0 and the calculated variance. Either way you will obtain the series of errors and applying the same idea as in the video will allow you to forecast ARIMA(p,d,q) manually. All the best!