Hi Jim,
Strictly speaking, a genetic chimera is an individual with a mixture of genetically distinct cells due to a transplant or the the early fusion of two fraternal twin embryos. Therefore, if someone receives a transplant of bone marrow or any other tissue, the donor tissue cells will always have different DNA from that of the recipient (unless the donor and recipient are identical twins). So yes, transplant recipients usually have to take immunosuppressants for the rest of their lives to prevent organ rejection.
I don't know of any way for a bone marrow transplant to cause a change in hair color. Some of the cells in the transplant are hematopoietic stem cells which can differentiate into different kinds of blood cells (platelets, etc.) but they can't become skin or hair cells, as far as I know. However, some of the chemotherapy drugs that usually accompany a bone marrow transplant could affect hair color or texture, so this might be where that particular myth comes from.
Chimerism does indeed make DNA testing more complicated. In the case of bone marrow, a DNA test from a blood sample would match the donor while a test from hair or saliva would match the recipient. The recipient would also have the blood type of the donor if they did not already match.
Individuals with congenital chimerism (the fusion of fraternal twin embryos) will be a mosaic of genetically distinct cells, so the results of a DNA test will vary depending on which tissues the test is done on. I have even read of a case where a woman was suspected of kidnapping her own children because a DNA test suggested that she could not be their mother. The reason was that the woman was a chimera and the cells that the DNA test was performed on were genetically distinct from her germline cells (those that produce eggs).