Can I transfer ownership of games purchased under one account (gamertag) to a another...

What you are talking about (License Transferring) and transferring a game to someone else's gamertag are two different things. When someone buys something off of the market place 2 licenses are issued

1. The Gamertag

2. The console the game was purchased.

Those are the only licenses that are issued. You can not transfer a license from one gamertag to another. What license transferring is, is when you transfer the license from one console to another, not from one gamertag to another.

It is very unfortunate what happened to your friend but the bottom line is your gamertag and the the console it was purchased on owns the license and there is no way to transfer that license to your friend. Unfortunately keanakalia was correct when she said
that he would have to buy the game under his gamertag, If he wants to own the game himself.

No. Digital copies are tied to the account that purchased them and you cannot transfer ownership to another account.

Hey there Fang! When content is purchased on the Xbox Live marketplace, it is bound to the account that purchased it as well as the console it was originally purchased on. You'll need to make sure that your sister purchased all of those DLCs on the console
where you're trying to play them. If she didn't, then you'll need to be logged into her account to have access to the DLCs.

The first thing I would check would be which console she originally bought the content on. Then, once you confirm that you should be able to access the content (by either being on the original purchasing console or being logged into her account), you'll need
to make sure that all the content is downloaded and installed. If you need to redowload it, you can use your download history, as Joey suggested. Here's the link again:

support.xbox.com/.../download-content-history.

There's no way to transfer ownership of content from one account to another. License transfers are used to transfer ownership from one console to another, not to transfer ownership from one account to another. For example: If you were to buy a new console and
wanted to transfer the ownership of your content from the old console to the new console, you would use a license transfer.

Hey there Toffee!

Just to be clear, license transfers are not meant to be used for sharing content. They work for transferring ownership from one console to another, not from one profile to another. For example: If your brother bought the Dragonborn DLC then
gets a new Xbox 360 console, he can use a license transfer to transfer ownership of the content to the new console. But it cannot be used to transfer ownership to your profile.

If you are wanting to play the DLC that he purchased on his profile, you will need to be logged into his profile to do so.

Are you logged into his account while trying to access the Dragonborn DLC?

Sorry, you cannot transfer games, history, purchases etc between accounts. You can change your gamertag, your email, your physical address associated with your account etc but you can't actually transfer between accounts. Everything you do under one account
will stay under that one account and cannot be changed.