Once you create an Xbox Live account, your gamertag replaces whatever name you gave the profile. Profile names are local to the system, but Xbox Live account gamertags are unique across all Xboxes, so it's unlikely that he could change it to be "Dave" again - likely someone already has that tag. I do believe you get one "free" gamertag change when you sign up for live (after that it costs points, which cost money), so if there's some gamertag that would suffice for his purposes, he might just be able to change it.
If you have an Xbox Live account and want to buy DLC, it will work fine on all profiles on the first Xbox it is downloaded on, and the profile that purchased the content on any Xbox.
The only downside here would be that if your situation changes, it might not be easy to use content you purchased previously. For instance, if you get your own Xbox and download the previously purchased content to it, it will only work on your profile/account, and not on any others on the new Xbox. There's a "license transfer" option in the dashboard that can get around this, although you'll then deauthorize the content on the first Xbox, and it can be done only once every four months. Needless to say, this gets tricky kind of fast.
For the purposes of free DLC, this caveat likely doesn't matter. You can use the new Silver account your dad created with the "random" gamertag to download free items off the store, and create a new profile with the name you prefer to play with.