Does all cancer make hair fall out?

Essentially, none do. Cancers do not cause hair loss. Treatments for cancer cause the hair to fall out. Chemotherapy involves giving chemical that kill fast-growing cells at certain phases of growth in all parts of the body, including normal cells. Hair follicles under the skin are where the hair stops growth when chemotherapy starts. Very fine hair pieces are available until normal hair reflows (many months).
No. It's not cancer that makes hair fall out but the treatment. Some chemotherapy and hair in the radiated fields cause hair to fall out. So not every patient who has cancer will have hair loss.
No. Generally it is not the cancer but the treatments that lead to hair loss. Some chemotherapy drugs can cause hair loss. Radiation treatments can cause hair loss if the brain or skull is treated.
No. It is usually certain types of treatment for cancer (chemotherapy) that causes loss of hair, not the cancer itself.