Model and data source
As in any other applications you should start from the plausible model, that would fit most of the countries in your cross-sectional dimension. Suppose you do have a quite universal theory that fits most of the countries in the world. Then you could try to include as many countries as possible using either World Bank, IFS statistics or simply Penn World table data. However you may also limit yourself to more homogeneous samples like EU data taken from Eurostat or OECD statistics.
Cross-section of countries
Since in panel data models (you may also consider linear mixed effects models as an option, since you also include interaction terms) under you (mostly) quantitative hypothesis "was there a significant (both statistically and economically) impact of belonging to euro-zone club on the budget deficit?" you have to balance the number of countries that are in the euro-zone, adopted it from a certain year, and are out of the club.
Should I only use countries that have introduced the Euro?
Is adding other European countries or western like countries beneficial as they add more data points and/or acts like a reference group?
Regarding these particular questions, in my opinion, the inclusion of EU countries and some OECD rivals to France, Germany and Italy, would be sufficient for your analysis. You have not to limit only to euro-zone club, because of selection bias. Well some countries are adopted euro within the time dimension, but it is better to compare with the countries that are not so restricting their fiscal policy due to strict Maastricht criteria they obliged to follow.
Or is id only bad for the data to add countries that has not introduced the euro as they have no variation in the variable of interest (EURODUMMY)?
Dummy is a variable that has two values $0$ and $1$, so if you include only (not in this case though) $1$ you will face pure multicollinearity problem with the common intercept term, that is present by default in