Question: Does it have anything to do with Lelouch and what happens after R2? I need to know whether or not they continue his story or did he become immortal from his father's code?

Answer: 
In a sense, it does relate to the 2 seasons but not as a continuation of R2, Akito of the Exiled is set between the 2 seasons of Lelouch of the Rebellion during the 1 year gap, in it Britannia is putting pressure on it's EU Front.

Now I haven't watched the series yet however from screenshots we see C.C in this wearing a white hood and a man who looks like Lelouch with Suzaku (having been made a Knight of the Round) going by a different name with one of his eyes covered. In episode 3 of the OVA series it's revealed that Julius Kingsley is infact a brainwashed Lelouch. The brainwashing however is somewhat unstable as "Lelouch" reappears briefly, also his Geass has yet to be sealed.

> 
>  ![Lelouch Reappeaering](https://i.stack.imgur.com/0M2Mt.jpg)
> 
> 
>
**NOTE:** The answer is accurate at the time of posting. In it, we disregard future events, such as the producers deciding to make another season or sequel.

---

The first part of your question was answered by Memor-X. "Akito the Exiled" is an intermission between the two seasons of the main show. 

The second part?

[The canon is that Lelouch is dead the end of R2](https://anime.stackexchange.com/a/2438/2808), period.

There is some speculation that he was the cart driver, that he became immortal, etc, but having [Word of God](http://tvtropes.org/pmwiki/pmwiki.php/Main/WordOfGod) that he is dead, all the rest is speculation even if they changed their minds after leaving lelouch's smiling at the cart bench in the cutting room's floor.

So, 

1. Yes, Akito the Exiled is in the same world as Code Geass, happens in Europe, chronologically between the two seasons of the main show.
2. No, he (Lelouch) did not inherit his father's code.
3. No, Lelouch did not survive being stabbed by Suzaku.
4. No, he (Lelouch) is not the cart driver, even though many of us wished for that.
5. No, they did not continue the main storyline after the end of R2.