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 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.
