We have stable and well fitted basic LS model for panel data where VIF for each variable is below 5 . However statistical tests indicate that fixed effect model is more appropriate. When individual effects are introduced VIF increases significantly. Is it a problem or regular feature of fixed models that we can ignore?
Countries' dummy variables are also independent variables and their correlation with other explanatory variables should not be ignored. I assume that some of your explanatory variables are time-invariant or almost time-invariant and that is why they are correlated with countries' fixed effects (which by definition are also time-invariant).
You can use time-invariant explanatory variables in the random effects specification or, even better, in the within-between specification which is comprehensively described in: Bell A and Jones K (2015) Explaining Fixed Effects: Random Effects Modeling of
Time-Series Cross-Sectional and Panel Data. Political Science Research and Methods 3(1): 133-153.