Since you are a student and presumably interested in publishing your results in some form (even if only as a thesis), take advantage of your student status and try to find someone locally with statistical expertise to help you. You thesis adviser should be able to point you to someone, even if your adviser doesn't have such expertise directly. That will be in your long-term interest.
I'll try to address your questions, in general, in the meantime.
First, with counts as the response/dependent variable, you are going to use some type of generalized linear model; standard linear regression is inappropriate. Whether you analyze this as Poisson or negative binomial, consider zero-inflated models, etc., depends to a great extent on the nature of your data. This Cross Validated page provides a good place to start examining the basis for such choices.
Second, the decision between fixed and random effects in this type of analysis, for countries in your case, generally depends on whether you care about each country individually or if you just want to have general corrections for possible between-country differences. If you want to examine countries individually as fixed effects, each country will use one degree of freedom in your analysis, while taking them as random effects only uses one degree of freedom for each variable adjusted for random effects.
For your particular data set, with 130 countries, only 10 years of data, and large numbers of 0 entries, you probably don't have enough data to treat countries as fixed effects. In general, the number of predictor variables you can analyze is limited by the number of counts. My guess is that you'd be better off with treating countries as random effects, which along with the fixed effects you care about by definition makes your analysis a mixed model.
Note that your work isn't done just by calling countries "random effects" in a "mixed model"; you have to decide whether the random effects just affect the baseline FDI (intercept term) or whether countries might differ in things like the relationship between FDI and GDP (slope terms). Your model needs to be specified carefully so that it accomplishes what you want. I find these specifications of random effects to be somewhat confusing, so I strongly recommend getting some help with someone who can go over your data and your analysis plans with you in detail.
Third, with respect to generalized linear models versus GEE, I'm not an expert but I can provide some general direction. As a practical matter, if you have missing data there are different underlying assumptions about the nature of the missing data in the SPSS analyses, with less stringent assumptions required for the generalized linear model. I also understand that GEE can more readily provide information about things like rates of change over time than a generalized linear model does, but I might be wrong on that. So the choice about using GEE depends on whether you need that extra flexibility to examine your specific hypotheses.
Overall, use this as an opportunity to learn about the strengths and weaknesses of each of these approaches. There may be no better way to learn than by trying out different types of analyses on a data set you care about, ideally with an experienced guide at your side.