1) When we omit the intercept, aren't we forcing the regression line through the origin? Does that pose any problem because we assume that there is no variable that affects the outcome other than the independent variables we have in the model? 2) Is it okay for the intercept to be significant in a regression model with categorical/dummy variables?
Q1: If you omit the intercept in univariate regression, you not only assume that no other (included) independent variable has an impact on the dependent variable – you also assume that the dependent variable is zero if the independent variable is zero. If the intercept of the true model is non-zero, doing OLS without the intercept induces potentially huge bias. It is in general preferable to include the coefficient for the intercept and see if it is significant, unless you are sure the specification really does not include the intercept.
Q2: Dummy variable is pretty much similar to any other independent variable. So the intercept may be significant in the presence of dummies. The issue is if all the observations had the same value for the dummy as then you would have perfect multicollinearity with the intercept – but again, this is an issue for any independent variable.