This sounds like it might be an appropriate situation for multilevel modeling. How many different regions do you have? If there are many (say, dozens or more) you might wish to take such an approach (c.f. Duncan et al., 1998)
On the other hand, educational attainment can be incorporated as a numerical predictor quite successfully, although I always explore its functional relationship with the outcome by (1) using a nonparametric smoothing regression (Beck, 1997; Hastie and Tibshirani, 1987) in order to inform (2) specify a nonlinear (in all likelihood) functional form, usually with nonlinear least squares regression (Davidson, 2004).
If there are relatively few political parties, you might wish to retain the indicator variables for these categories.
References
Beck, N. and Jackman, S. (1997). Getting the mean right is a good thing: gen- eralized additive models. Working paper, Society for Political Methodology.
Davidson, R. and MacKinnon, J. G. (2004). Econometric Theory and Methods, chapter 6: Nonlinear Regression. New York: Oxford University Press.
Duncan, C., Jones, K., and Moon, G. (1998). Context, composition and heterogeneity: Using multilevel models in health research. Social Science & Medicine, 46(1):97–117.
Hastie, T. and Tibshirani, R. (1987). Generalized Additive Models: Some Applications. Journal of the American Statistical Association, 82(398):371–386.
$b_3$ is the difference between white females and the sum of $a+b_1+b_2$. That is, the difference between white females and the sum of non-white males plus the difference between non-white females and non-white males plus the difference between white males and non-white males.
\begin{align}
b_3 = \bar x_\text{white female} - \big[&\ \ \bar x_\text{non-white male}\quad\quad\quad\quad\quad\quad\quad\ \ + \\
&(\bar x_\text{non-white female} - \bar x_\text{non-white male}) + \\
&(\bar x_\text{white male}\quad\quad\! - \bar x_\text{non-white male})\quad\ \big]
\end{align}
Honestly, it's a bit of a mess to interpret in this way. More typically, we interpret the test of $b_3$ as a test of the additivity of the effects of ${\rm white}$ and ${\rm female}$. (The expression within the square brackets $[]$ is the additive effect of ${\rm white}$ and ${\rm female}$.) Then we make more substantive interpretations only of simple effects (i.e., the effect of one factor within a pre-specified level of the other factor). People rarely try to interpret the interaction effect / coefficient in isolation.
It may also help you to read my answer here: Interpretation of betas when there are multiple categorical variables, which covers an analogous, but simpler, situation without the interaction.
Best Answer
The multiplication scheme only works if you want to treat the education variable as a continuous or ordinal variable that is linearly related to your dependent variable after controlling for sex. In most of the cases (from my experience), this linear assumption seldom holds as there are just too many heterogeneity within some of the educational categories.
If you treat education as a categorical variable, the computation of interaction terms is a bit tricky. Generally, if you have two categorical variables: $x_1$ with $j$ levels and $x_2$ with $k$ levels, to completely model their interactions you'll need $(j-1)\times (k-1)$ dummies. Here are the possible schemes:
Variable $female$ has two levels and variable $education$ has three, so to model the interaction you'll need $(2-1)\times(3-1) = 2$ more dummies on top of the dummies used for main effects.
For instance, if people in college is your reference group, to model the main effect, you'd need $female, D_{Elementary}, D_{Middle}$. To further model the interaction, you'll then need add the products $female\times D_{Elementary}$ and $female\times D_{Middle}$, which is Scheme 1 in the table.
Alternately, if you use other levels in your education variable as reference group, you can change your scheme accordingly. But overall, you should have 5 binary independent variables. These five dummies and the intercept together will allow you to estimate all the 6 means (2 sexes by 3 education levels = 6 possible combinations).
In real setting, we rarely do that by hands. Most software packages allow us to assign the type of variables so that the regression analysis will handle the variable appropriately.
In SAS, look into
classstatement inproc glm; in SPSS, check thefactorandcovariatepanel in glm module; in R, usefactor()oras.factor()functions to change the variable's nature; in Stata, look into adding prefixi.before your independent variable.