First differences vs. fixed effects model for panel data I'm aware of the fact that first differences and fixed effects are both designed for the same solution -- removing unobserved unit-level effects.
However, I'm unclear on what happens when you include a unit-level dummy in a first differences model (I've seen this done for error-correction models as well as elsewhere). Does including the dummy essentially demean the differences (sort of a soft unit-level time trend?). Or does it simply introduce bias?
 A: If you have $N$ individuals and you include $N-1$ individual dummies (one less in order to avoid the dummy variable trap) in an OLS regression like
$$y_{it} = X'_{it}\beta + \sum_{i=1}^{N-1}\delta_i (\text{individual}_i) + \epsilon_{it}$$
then this is called a least squares dummy variable (LSDV) regression. In this case, each individual dummy will "absorb" the individual fixed effects $u_i$ that are hidden in the error term $\epsilon_{it} = u_i + e_{it}$.
Mundlak (1978) has shown that the LSDV regression is equivalent to the fixed effects estimator:
$$y_{it} - \overline{y}_{i} = (X_{it} - \overline{X}_i)\beta + \epsilon_{it} - \overline{\epsilon}_i$$
where $\overline{y}_{i} = \frac{1}{T}\sum^{T}_{t=1}y_{it}$, $\overline{x}_{i} = \frac{1}{T}\sum^{T}_{t=1}x_{it}$, and $\overline{\epsilon}_{i} = \frac{1}{T}\sum^{T}_{t=1}\epsilon_{it}$. Back in the days when computers weren't very fast, having large panels basically made LSDV infeasible because there were too many dummies. Therefore Mundlak's finding was very useful because it dispenses of including all these individual dummies and instead using the within transformation made things much simpler.
So if you do a fixed effects regression you don't need to include all individual dummies. In fact, your statistical software will just drop them should you include them in a fixed effects regression. Also in a first differences regression the individual dummies will drop out because they do not change over time, hence the difference is zero for all the dummies and then your statistical software will omit them due to perfect collinearity. Doing either fixed effects or first differences already solves the problem of time-invariant unobserved variables ($u_i$). LSDV is just another way of doing it and for this reason it won't help you to combine it with the other methods.
When you include individual dummies after first differencing your other variables, i.e. a first differences regression with individual dummies, those dummies will estimate individual trend effects (see page 77, footnote 1 in the notes here).
