I have Ultimate Edition of New Vegas. My friend has the regular edition with none of the DLCs. If I were to give him the DLC disc and he installs it on his xbox, would he be able to play the DLCs with the regular disc or would he need the Ultimate Edition play disc as well? I also have the regular edition disc from before I bought the UE and kept it as a Backup disc.

It will work. I have the ultimate edition and my friend has the standard edition, we did this and it worked fine

The DLCs save to your hard drive so you only need a normal disc to play it.

I can confirm this. A friend of a friend lent my friend a copy of F3 GOTY to install the DLCs while only having the base game. It worked.