Do blood cells immediately die after leaving the body?
No. If they did, blood transfusions wouldn't be possible (or they would require some sort of direct body-to-body system).

Red cells collected can be stored under refrigeration for over a month. See for example the description of what happens to donated blood from the Red Cross. Platelets don't last quite as long, but they also survive for some time in the right conditions. Of course, some cells die during storage, but there is no instantaneous die-off once cells leave the body.