What are the advantages and disadvantages of including a baseline measurement variable in longitudinal data analysis? What are the advantages and disadvantages of considering a baseline measurement variable in longitudinal data analysis? (A baseline variable could be the number of seizures each patient had per measured day before the study began.)
In a randomized-control trial, does the baseline variable have to be before the treatment began, or could it be on day one of when the start of treatments began?
This sounds a lot like feature selection.
 A: In general, having a baseline measurement will afford greater power for the analyses of interest.  In addition, it may make the model's output correspond more closely to the substantive research question (e.g., how do patients change as a response to treatment vs. how does one group differ from another).
In biomedical research (i.e., clinical trials), the baseline measurement is prior to treatment.  That said, it could be the same day as treatment, it just has to be prior.

I don't understand the point about feature selection.  One interpretation of your question is if the baseline should be a separate covariate, or if it should be the $y$-value at ${\rm time}$ $0$ in a longitudinal dataset where there are multiple $y$-values for each statistical unit (patient) associated with different values of a ${\rm time}$ variable.  If that's what you're getting at, this is a generalization of the longstanding debate about whether it is better to model data using change scores or ANCOVA.  See:

*

*Best practice when analysing pre-post treatment-control designs

*Is it valid to include a baseline measure as control variable when testing the effect of an independent variable on change scores?

*Should the difference between control and treatment be modelled explicitly or implicitly?
(among others...)

