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A factor would be a categorical variable, whereas a covariate would be a continuous variable.

And comparing group1:age and group2:age would be testing for the differences in age effect between group1 and group2?

If they're not, then leave them out, because they don't seem to provide any extra benefit with respect to more DE genes or lower dispersion estimates.

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Lastly, if I were to include another continuous variable(s) as covariates into the design matrix such that: in order to adjust for the covariates continuous1 and continuous2, other than being adjusted for them, would they affect how the above comparisons turn out. I am quite confused by the usage of design matrix, especially with continuous variables interaction. So, if you drop the second coefficient, you'll be testing for the same intercept between groups 1 and 2, i.e., your null hypothesis is that groups 1 and 2 have the same expression at an age of zero.

If there is any confusion regarding the questions I ask, please tell me. This doesn't really seem very sensible to me - babies aside, would the behaviour of zero-age samples be particularly interesting? The first contrast tests for a differential response to age between groups 1 and 2, while the second contrast tests for any differences in expression between groups 1 and 2 (as the first contrast will not reject the null hypothesis if the gradient is the same between groups). In general, I'd expect a decrease in the sample variance as you're increasing the complexity of the model to account for more sources of variability.