So, I obtained a DOD fellowship during my PhD and there were no major problems with it. It was not the NDSEG but was instead through the Breast Cancer Research Program (
http://cdmrp.army.mil/bcrp/). I believe the procedure should be pretty much the same though. Probably this NDSEG grant is a little more based on the individual (like NSF fellowships) while the one I did was based on a specific research proposal. Doesn't really matter.
Pretty much all these fellowships work the same way. If you get one, it can pay your salary and your tuition. If they allow for a smaller stipend than your other MD/PhD colleagues, the program or your PhD advisor will usually make up the difference. You often can't use these fellowhips during your M1 or M2 because you need to be actively pursuing your PhD work. This makes the best time to apply during your M2 or your first year of full time PhD work. While you are receiving the money from the fellowship, you are not paid by your advisor or MSTP program.
If you are in an MSTP program where your funding is fully provided the entire time (either by the MSTP grant or your PhD advisor), then it's not likely to make too much of a difference to you personally. It will give you a certain amount of prestige and make you more desirable as a PhD student (for your advisor or potential advisor). However, it will save either your program or advisor some money. Also, should certain situations arise (like your PI deciding to leave the institution), you gain a major advantage by having your own funding.
This works the same way for almost every fellowship: AHA, DOD, Hertz foundation, whatever. The only thing you should be aware of is that MD/PhD students are not eligible to get certain fellowships (NSF, for instance).
