The assumption is typically phrased in terms of the error variance, but the implication is there for standard deviation.

1. Your interpretation is correct. Remember, though, that this means for every possible combination of the values of the independent variables (male 55 years old, woman 55 years old, male 56 years old, female 56 years old, and so on, assuming only gender and age were in the model). As you can tell, that gets more tricky to think about as we add more independent variables. This can be checked (informally, but quite effectively) by creating a scatter plot of the regression residuals (as an estimate of errors, on the Y-axis) against the respective, predicted values of Y (X-axis). This allows us to look at the assumption in two dimensions (using predicted Y-values is a way to reduce the dimensions that arise from having more than one independent variable). You should expect to see no discernible pattern in the residuals as the predicted y-values change (implicitly, as the combinations of x-values change). Some classic violation patterns are cones/triangles/football/bullet patterns in the residuals.

2. In theory, it should be the same, but in practice you'll never have it identical due to sampling variation. In general, you examine the assumption and hope you don't see any evidence that it's violated (i.e. too different).