What's the difference between observable fixed effect and control variable? I am confused about the exact definitions here.
Assuming I have a cross-sectional regression, let's say, Wage on Education and I additionally control for observable characteristics with a set of dummies or variables like intelligence level, age, parent's education level, urbanization area, gender, race, work experience etc. 


*

*Does this mean I used dummies/variables to "control for observable fixed effects" that I obtained through my data collection? (Is, for example, parent's education level thus an observable fixed effect?)

*Do unobservable fixed effects like ability (as often quoted in the literature) are then said to be "controlled" by proxies through my dummies like intelligence level, experience etc.?
In a way I'd like to know the exact difference between controlling for a variable, observable and unobservable characteristics, observable and unobservable fixed effects. Thanks.
 A: Using your example for your first question: Yes, parent's education level is a fixed effect as you set them to be at any arbitrary number you can/want to use. There is no "randomness" regarding the levels, you fix them, you fully observe them. 
Regarding your second question: The Nunn & Wantchekon article is a good clarification for what you want. I am no expert on the matter but I think your intuition is correct. Ultimately it is an OLS; the way they test for significance is somewhat specialized but otherwise nothing crazy. So... "Yes" is also your second question's answer. Ν. & W. strive to have a somewhat rigorous falsification of their findings, checking of their "unknown unknowns" but yes, otherwise nothing too exotic. In general, if there is an observable fixed effects you can include it in the OLS to start with. If you suspect there are some unobservable ones you try to control for them by adding dummies; surrogate variables if you like. 
As mentioned, I am not an expert on panel data and I haven't work in economics ever so take it with a grain of salt. I believe your statistical intuitions are not wrong though.
