Don’t panic too much about the sample storage and contamination. You won’t need the cells alive for the DNA test so that makes it easier. Also the quick turn around (i.e getting it to the person carrying out the tests will mean they don’t even really have to be kept on ice, although granted that would be preferable). It should be ok. If you need any advice on the science bit I can help as I’ve worked in labs for about 7 years (I should really know what I’m doing by now! lol).
Anyways, try not to worry too much, what you are doing is for the good of people who can’t help themselves at the moment.