What is the risk of serious bacterial infection in a febrile solid tumor patient who has not yet started cancer-directed therapy and has normal cell counts with no central line?  

This depends on the type of the tumor, the location, and the individual circumstances. It may not be wrong in this scenario to draw blood cultures and give a dose of ceftriaxone pending 24-hour results, but this practice may also vary based on these different variables e.g. RMS of paranasal-ear area...