First question: is it possible for a multiple regression model to have "big" and significant coefficients but a low $R^2$ value?
Let's say the value of $R^2$ is 0.0005 and my coefficient of interest is 0.6, ignoring the coefficients of all other variables I adjust for. And the dependent variable has
range = [0,5]
mean = 2.5
variance = 4
Second question: if that's possible, does the low $R^2$ value affect the validity and the interpretation of my coefficient of interest?
Edit: I might be using the term validity loosely here. But my understanding is $R^2$ quantifies how much variation in the dependent variable explained by the model. So, does it make sense that I interpret the effect of the coefficient as usual (e.g. holding other things constant, 1 unit increase in my variable of interest increases the value of dependent variable by 0.6 on average) even though it explains very little of variation in the dependent variable?