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?

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?

