Regression AR(p) models and stationarity

I am starting to learn time series models besides the expoential smoothing ones and I got a few questions that I am struggling with.

1. If I have a stationary time series wich follows an AR(1) process, should I get the same results using either AR (1) model or a linear regression with a explanatory variable equal to a lag version of the time series (1 in this case). Regarding p values, would them be likely similar (assuming AR model might fit using something different the OLS).

2. If I have a non stationary series and I do the fit using again the regression with the lagged value and the AR(1) process, would them be equal or now non stationarity, would make differences in the estimated parameters.

3. I keep reading about using ARMA models only for stationary series, what is the problem if the time series is not stationary?, I´ve simulated some simple non stationary AR(1) model, and the estimated parameters seems quite precise regarding the true process. Is it that estimated parameters might differ from the real process ones when an AR(p1) is used (asumming thats the real process), or p values might become not trusty anymore ? (I know I can check stationarity and unit root testing, but I want to know if even ignoring it, I might get good results like the simulated one I did and why that might be).

Thanks in advance, finally if u can sugest a book to get some of this answers would be super helpful also.