Yes it is perfectly possible. You need to include the interaction term into your model. The type of the model will depend on the type of dependent variable and your hypothesis.
Indeed. say that one factor is gender being female (D1) and the other is having a University degree (D2) and your response variable is wage (y) presumably in logs. Then you could of course estimate the model y=B1*D1+B2*D2+e (where e is an error term) and get the main effects that is the gender and the University effect on wages, but you may equally well estmate y=B1*D1+B2*D2+B3*D1*D2+e where B3 would estimate the additional effect of University degree for females on wages. This is readily done in a regression framework although in this simple example it is sufficient to compute the mean wages in the four groups in your sample, namely males/females and w/wo University degree.
Yes - it is common to run this kind of model as an analysis of variance (ANOVA), though this is just a special case of multiple regression (usually with a slightly different parameterisation). Depending on software running it as an ANOVA may offer a wider range of options.
You can use regression model or anova model depends on your experimental design.
How do you interpret it?
Bambor et al. (2006) says that you should interpret the conditional effect, or the impact of D1/y=beta1+b3(meanD2). And in this case D2 is categorical = 1 or 0 so the impact of D1 on y when D2=1 is beta1+beta3.
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