Interpreting Interaction Variables With Numerous Variables I have a model that has the following results: 
Is it possible for me to interpret a white man who received Degree1 and went into Profession1 as having a wage as $15,230.05 ($12,917.22 + $357.53 + $2,319.28 -$363.98)? Or would I have to have the gender variable apart of the interaction, i.e. WomenDegree1Profession1? If so, how do I interpret the model as is?
 A: You seem to have the basics down OK, if I understand the coding of the predictors correctly and the model actually represents what you want. I assume that this is standard treatment coding with reference levels of male for sex, white for race, some Degree0 for Degree and some Profession0 for Profession.
Then for your example, you would start with the intercept, wouldn't have to add anything for sex or race, and then add the terms for Degree1, Profession1, and their interaction. That's what you did.
Now for the warnings.
First, I don't see in your display the expected interaction terms for Degree1:Profession2 or Degree2:Profession1. If you set up the model without those interaction terms it probably can't be interpreted very well at all. It's generally important to include all lower-level and interaction terms together in the model.
Second, you ask "would I have to have the gender variable a part of the interaction"? That's really up to you. The model as you show it implicitly assumes that all effects of Profession and Degree and Race and the Degree:Profession interaction are the same for both men and women. If that agrees with your conscious assumptions, OK. If you think that associations of any of the other predictors with outcome differ depending on sex, then you need to include interactions with sex.
Third, you need to be able to evaluate the standard errors around the point estimates that you make. For that you need the covariance matrix of the predictors and the formula for the variance of a weighted sum. It's less error-prone to ask the statistical software to do all of these calculation for you.
A: You should rethink your model.
Here are some points to consider:

*

*Gender has only a main effect and it's positive for women. This means that for any combination of degree and profession (and race), women can expect to get paid \$3,965 more than men. This outcome doesn't align well with what research and the news tell us about the gender pay gap. Take for example the "base" salary with degree 0 in profession 0. In that case (white) men get paid \$12,917 and (white) women \$12,917 + \$3,965 = \$16,882 or 30% more than men. This result is hard to interpret/explain though of course full understanding depends on what profession 0 actually is. [Frankly, it will make more sense if there is a typo and women get paid \$3,964 less than men on average.]

*The main effect for race is a bit hard to take at face value as well, as it says that the salary for people of color is only \$501 less (independent of educational attainment and choice of profession).

*In short, since Gender and Race don't interact with Degree and Profession, the model assumes that the gender pay gap and the race pay gap -- if they exist -- don't vary with education and profession. It's a very strong assumption.

*You include a Degree-Profession interaction but it's incomplete. As @EdM points out, since both Gender and Profession have three levels (a reference level, level 1 and level 2), we would expect 4 interaction terms: Degree1:Profession1, Degree2:Profession1, Degree1:Profession2, Degree2:Profession2. Obviously, you have explicitly excluded two interaction terms; why?

