I have no expertise for your first question. For the second one the answer is no. Taking a 12-month seasonal difference will generally produce a series with seasonality. Typivcally, there will be negative autocorrelation at lag 12. That explains the success of the so-called airline model (a mixed non-sesonal and seasonal moving average on the 12-month seasonal difference, often on logged data).

Hi, my answers to your questions:

1. No, it is not compulsory but...if you do not do something to deal with the seasonal behavior of your data, your estimates will be reflecting two effects: those coming from common seasonality and any other effect that is not seasonal and that could be linked to your tourists arrivals. So, if you want to get a deeper understanding of the spillovers in tourist arrivals, beyond those contained in seasonal effects, you will have to deal with this seasonality in some way: one way is to estimate your VAR with and without seasonal adjustment in your data. That will help to see how the seasonal patterns affect your results. Another avenue would be to include seasonal dummies as exogenous variables in your VAR. I would suggest to follow all these three approaches to have a more comprehensive view of what is going on.

2. Taking 12-month seasonal differences is one of the simplest and most intuitive ways to try remove the seasonal pattern in your data. Sometimes it helps a lot, some other times it does not help. Let me explain one of the problems with this approach. Most of our times series can be thought of the sum of at least two components: a seasonal componen s(t) and a non seasonal component x(t). So lets say that your series looks like this: y(t)=x(t)+s(t). Once you take the 12 month difference you have y(t)-y(t-12)=x(t)-x(t-12)+s(t)-s(t-12). The good thing is that now you have removed the seasonal component from s(t)-s(t-12), nevertheless, the cost you pay is to GENERATE a previously nonexistent seasonal component in the term x(t)-x(t-12). So, you try to solve a problem just to generate another one.

I agree with the previous response, this is probably one of the reasons why the airline model does a good job when forecasting series in 12 month differences.

Hope it helps!