Not necessarily. Blood yes, saliva is not required to go into hazardous.
No. Medical waste regulated at state level with osha guidance. Most states require the following items to be placed in "biohazard" containers: -sharps -microbiology lab cultures -pathologic waste /human tissues -items from isolation rooms -blood or other potentially infective liquids and items "saturated" with these fluids. See your state doh and osha blood born pathogen standard.