4. Forever contain both DNAs
Yes, the donor kidney still has (and will remain to have) the donor's DNA. That is irrelevant to the acceptance and continued functioning of the organ, though.
The thing that could make an organ transplant fail is the recipient's immune system. If it detects the organ as "alien", it will attack it like any other foreign particle, and the body will reject the transplant.
To avoid this, two things are being done:
A donor is sought whose organs have similar protein markers to those of the recipient. (Those markers, and not the DNA, are what the immune system looks for when trying to tell friend and foe apart. DNA is inside the cell, the protein markers are on the outside.)
Similar protein markers are usually found in people with similar DNA, which is why close relatives are tested for this kind of compatibility first.
The recipient recieves special drugs that partially suppress the immune system.
If the recipient's body does not reject the organ, it can continue to function normally despite its different DNA. If, for example, there is a genetic defect in there that will lead to some kidney-related issues at a later time, it will be the donor's defect. If there has been a defect in the recipients kidneys that would have led to issues later in his life, that defect would be gone with his kidney. Other cells in his body would still carry the same defect, but those cells are not kidney cells, and thus that particular issue will not manifest itself.
Replication of cells will follow the same pattern: Cells from the recipient will replicate the recipient's DNA, cells from the donor's organ will replicate the donor's DNA.