So my research question is: is there any factors that predicts whether you injure your dominant or non-dominant shoulder?
So a list of possible independent variables are gender (male/female), presence of diabetes (yes/no), injections (yes/no) etc.
So how do I work out the relationship of multiple independent variables combined to the dependent variable? I have worked out their individual relationships. What statistical tests should I be using?
Linear regression is what you need. Specifically, you are looking to do regression with multiple categorical variables. ANOVA, or analysis of variance, will help you better understand the influence/effect of each independent variables on the dependent variable.