Your first question: Close but not quite: you either need to leave one of your species out of the model as a reference category or you need to leave out the constant. In the former case each βi+2 measures the difference between species i and the reference species, so the parameter of the reference species is necessarily 0, which means that the indicator variable for the reference species drops out of the model. In the latter case each βi+2 is the constant for each species, which means that there is nothing left to do for the overall constant β0 so it should drop out.

Your second question: No, all the information is already captured by the species indicator variables (a term I prefer over dummy variable), so there is nothing for the family indicator variables left to explain and they will be automatically dropped from the model due to perfect multicolinearity.

So species indicator variables can be used to reflect a 2-level structure, but they do not reflect the additional structure of a 3- or 4-level model - am I reading this correctly? –  Roland

I don't understand that comment. I see only two levels: species and families. Where is that 3rd and 4th level comming from? As long as the levels are hierarchical, than indicator variables at the lowest level (e.g. species) will absorbe all the variance of all the higher levels (e.g. families). –  Maarten Buis

       
Oh, I might be using terminology incorrectly. With 2 levels I mean the individual pet level (level 1), and the species level (level 2): I'd call the simple linear model without species indicators a 1-level model, and as soon as we have species information, we would have a 2-level model. The third level would be one which includes families as well. A fourth level (order) was not written down in the models, but mentioned in the introduction. –  Roland

So family and order indicator variables will add nothing once you have included the species indicator variable. As a consequence, these will result in perfect colinearity and they will be dropped. –  Maarten Buis