You can have two different Xbox LIVE accounts on the same Xbox 360 for sure. I've had 4-5 on mine at one time between roommates and such.
You can also recover your Xbox LIVE Gamertag to any console at any point. Your account can only be "active and updated" on a SINGLE console at a time. (What this means is that it must be recovered to a console before you can use online services on it. This makes it so more than one person can't access your account at once.)
The way that purchasing marketplace items works is this:
Whenever you buy any item on the marketplace, it is tied to both your gamertag and your console. Let me give you an example.
Your gamertag is PersonA and your friend's gamertag is PersonB. Let's also say you have a buddy named PersonC who is visiting and has his Xbox LIVE account on a thumb drive.
Your Xbox 360 will be XboxA and each respective person's Xbox will be XboxB and XboxC.
Following along?
You buy an Arcade game on XboxA as PersonA. This means that ANY PERSON (A,B,C,etc.) can play this game on XboxA (YOUR Xbox), OR you can download this game to play it on XboxB or XboxC on your account only.
Make sense?
To summarize--when you purchase the game as PersonB on XboxC (just a random example), either PersonB can play it on any Xbox, or any person can play it on XboxC.
If this doesn't make sense I can clarify further.
Hope it helps! Happy gaming. :)
Edit: I wanted to add a tip. Say you and a buddy both own Halo: Reach and you want some new DLC for 800 MSP, right? Well, if one of you knows you will always be using the game on your own name on your Xbox and no one else will be, you can buy it on his Xbox. That way, anyone on his xbox can play it (him) and you can play it anywhere (on your own Xbox).
It's not illegal or cheating the system or anything, since the permissions are still restricted as such.
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.
Best Answer
I was able to "sign-in" to my own profile on my friends Xbox, and was able to perform the following:
By signing in on my account and proceeding to Settings -> Account -> Download History:
Redownload any games I had purchased from Xbox Live Arcade, even if my friend did not have them purchased himself. These were fully operable games, however, if my profile was not signed in, they were inaccessible, either with an error message stating the correct profile needed to be signed in, or by reverting to a trial version. (Games tested were Halo: Reach, Super Puzzle Fighter HD Turbo, Metal Slug XX, and King of Fighers 2002UM)
Redownload any DLC for games either I had purchased from the marketplace, either for games my friend already owned, or for games that I owned and had downloaded to my games. The same restrictions applied about my account being signed in. (DLC tested was the Leona character unock for Metal Slug XX, and the Halo: Reach Defiant map pack)
By being signed into my account, I was also able to access my cloud saves when opening a game, and selecting the appropriate device from which to load my saves at the prompt for load location. (Cloud saves tested were for Gears of War: Judgement)