Can the individuals who have downvoted my answer articulate why they did so? They may have very good reasons for doing so. And, in such case they should share such reasons.

As you know, I am not in favor of detrending none. Yet, it can result in an acceptable Cointegration model structure. Of course, you have to insure that such variables are cointegrated... which is rarely the case. However, there is one model structure where you can have a de-trended Y variable with a mix of de-trended and level independent variables... and that is in Error Correction Models (ECM).

No, I don't think so. Since you are dealing with annual data, from a seasonality and autocorrelation standpoint, only the first lag has true and independent meaning. The other lags (greater than 1) are just a function of the strength of the first lag. So, use ADF with just 1 lag. Your ADF test comes out with a high Tau stat and low p-value. So, you can reject the null hypothesis that this variable is non-stationary. You are in good shape.

I have tested variable selection between standard stepwise algorithms and LASSO. And, LASSO comes in a very distant second. LASSO penalizes variable influences, it can select weak variables over stronger variables. It can even cause variables signs to change. And, it breaks down the entire framework of statistical significance, Confidence Intervals, and Prediction Intervals. LASSO can at times work. But, look very carefully at the MSEs vs. Lambda graph and the Coefficients vs. Lambda graphs. That's where you can visually observe if your LASSO model worked.

Those are good clarifications. First, add just the dummy in your model, and see how your model improves. The model may be ok after that first step. If necessary, move on to the next step that we both describe. I call it "interaction variables", you call it a dummy "that interact with the independent variables". That is the same thing. You don't need to create interaction variables for all Xs. You could do it with just the main X variable to begin with. But, never create such a variable with the Y dependent variable on the right side of your regression. That won't work.