I personally think that Lelouch is dead. Why would C.C. give him her immortality? That would be a curse unto itself. He wanted to make her happy before she died. C.C. could have died along with Charles and Marianne if she wanted to. Then the entire Code Geass curse could have ended right there and then. But she chose to live on.
She chose to live on because, as you said, she wanted to be happy before she died. There are varying ways of looking at it but the one thing that runs through most of them are that (a) C.C. did want to die but more than that she wanted to be loved, or to belong (b) the Geass is a pact, you could see it as Lelouch helping her to die and giving her the love she wanted by taking the curse onto himself. She helped him remake the world, now he's fulfilled her wish. The second point also links into why she didn't want to die in the Ragnarok Connection; she hadn't fulfilled her pact with Lelouch yet and, as was said in the show, a Geass is like a promise, so you get the hard fact of the Geass mixed with the more romantic element that they wanted to help each other.
There's also the fact that the cart driver was in disguise. Now, the real life reason for that is to make in ambiguous. However, the only applicable in-universe reason for the driver being covered up is that he didn't want to be recognised. I can't think of any other character who would (a) not want to be recognised and would (b) be with C.C. in that situation.
Finally, it's true that being immortal is a curse but not only could Lelouch give it away when he's ready to die (and the recipient could do that same, and so on through history) but we also know that Lelouch is perfectly willing to suffer for what he thinks is right. I don't think he'd have much probably being "immortal" until he gives the curse to someone else when we'd just see him willingly get stabbed through the chest in order to realise his ideal. As I said, he suffered massively to get his part of the deal, now - potentially - he'll suffer to fulfil C.C.'s part of the deal; the return of her mortality. It also has a nice romantic element to it because he's showing his love by taking the curse and love is the other thing that C.C. wanted, as I said.