I usually collect samples in ice cold PBS. There is approx 30 mins time lag between the collection and the storage of samples. I keep samples in RNAlater at -20C for long term storage. Does this delay affects RNA quality ?
There are ways of store tissues for different applications.
For RNA work, tissues need to be a bit more careful. Recommended to snap freezing using liquid nitrogen or dry ice soon the tissues taken out, and keep them at -80 degrees for long term storage.
It depends on what you use the material for. Your wash is mainly to get rid of red blood cells supposely, If your target material is used only for RNA/DNA, it might not necessary to wash the red blood cells, as red blood cells are anucleate. If you are to looking for the cancer cell expression with the same material to analyse both RNA and protein, then you 'd better to get rid of as many of the red blood cells as possible by washing.