Any specimen that is going to be for display, is pretty much just skin and feathers on a stuffed framework. It will have likely been treated with harsh chemicals in order to preserve it, which will destroy any DNA beyond use.

I think the OP means just the DNA samples of the species alone, and not contained within a human genome.

It’s not just when the two species died, it’s also a factor of where/how they died. Mammoths are from cold arctic regions, and many of them died and have been encased in permafrost, which is essentially a natural freezer which helps preserve any samples.

One of the main issues with cancer, is that it is your own cells that are going haywire, so your immune system isn’t going to attack yourself and the cells replicate unchecked. If you introduce a cancer cell from another body, it will be recognized as foreign and initiate an immune response. It would be far more difficult to grow as it would be a foreign cell.