Let's say I were to impute values for a variable (using multiple imputation). Then I wanted to use that variable in a regression. Can I use the same variables I used to impute in my new regression? So, for example, if I were to impute math test score with the variables race and income, could I then regress on GPA the variables math test score (imputed), race, and income?
There is inherently no reason that one cannot use the same variables in the imputation model and the analysis model. In fact, one will typically want to put as many or more information into the imputation.
The only possible concern I could see is that if your imputation model is insufficiently complex and you are imputing a considerable proportion of your data, then what you assumed int he imputation may turn up in your analysis again.
For example, the "standard" MI model might be seen as one, where multivariate normality is assumed with some means and covariance matrix to be estimated - this essentially corresponds to linear relationships between the variables you put in (unless you make sure that you do something else). If most of your data is imputed, you should then not be surprised if the relatioship between two variables looks pretty linear, if you did not allow the imputation model to consider a non-linear relationship.
Imputation does not create bias in regression but it does underestimate variability. That is why multiple imputation is valuable. It accounts for the variability that is missing from simple imputation.
Keep in mind that it rests on the assumption that the missing values are missing at random (MAR). Read Rubin's book Multiple Imputation for Non response in Surveys and/or Little and Rubin's text Statistical Analysis with Missing Data.