Hello! I am trying to understand the different Panel Data models and I am getting confused by the different terms used, i.e., Random effect models and Random effects estimators and Fixed effect models and Fixed effect estimators- are these 4 all different?

Is it the case that you can have a fixed effect estimator in a random effects model and if you do that your estimate is not consistent?

For context, I am running a panel data analysis in STATA and after conducting a Hausman Test I obtain Prob > chi2 = 0.4504, indicating I should use a random effects model. However, I strongly doubt that my individual effects are uncorrelated with my x variables. Should I go ahead with FE or follow what is recommended? I have seen that a command, sigmamore can also be added to the test as it is less likely to produce a non–positive-definite-differenced covariance matrix, but how do I check if I need this?

Also, does RE model being the best-suited model mean I simply use the ,re output as my result or does it mean there's something in my variables and controls that needs to be changed or considered?

I am running a regression of total inflow migration on gov spending on healthcare per capita. My other x variables are population density, share of elderly, total tax revenue, GDP per capita and Gini.

I understand that I am not to include variables that do not vary over time within a country- or is that only for fixed effects? So I'm not sure if my x variables are 'correct'

I'm sorry these are quite a few questions but any help at all would be GREATLY appreciated. The more that I try to read about panel data models, the more I get myself confused so if someone could explain my specific case that would be amazing. Thank you so so much!!