RDD, wages, and day of birth
(1) Since the RD design identifies an effect only at the cutoff (in your case, at precisely 12am on the last day of the year), I don't think it's an issue of data manipulation for you to be removing the tails. Your estimates really shouldn't be sensitive to observations far away from the cutoff, but this depends somewhat on how granular your data is and how many observations you have (a lot of birth data is quarterly, which may make this strategy difficult).

(2) Sharp vs. fuzzy design refers to the assignment of the treatment. If your treatment of interest is being born December 31st vs. being born January 1st, then this is a sharp design because the running variable deterministically assigns treatment. If there is noncompliance with treatment, (i.e you were born December 30 but you somehow end up getting "treated" as though you were born in January the following year) this is when you would use the fuzzy design.

(3) Sharp RD is just a special case of fuzzy RD. Say $Y$ is your outcome of interest (in this case, wages), $X$ is your running variable (day of the year), $D$ is a dummy variable for treatment assignment where $D=1$ if $X&gt;c$ and $D=0$ if $X\leq c$, and $c$ is the cutoff (December 31). Then the treatment effect for the fuzzy design is (from Lee and Lemieux 2010):

$$
\tau_{fuzzy} = \dfrac{\lim_{\varepsilon\downarrow0} \mathbb{E} \left[ Y | X = c + \varepsilon \right] - \lim_{\varepsilon\uparrow0} \mathbb{E} \left[ Y | X = c + \varepsilon  \right]}{\lim_{\varepsilon\downarrow0} \mathbb{E} \left[ D | X = c + \varepsilon  \right] - \lim_{\varepsilon\uparrow0} \mathbb{E} \left[ D | X = c + \varepsilon  \right]}
$$

Because $X&gt;c$ means by definition $D=1$, the denominator reduces to $1$, leaving us with the sharp treatment effect.