Yeah, Duryodhan named both his children Lakshman(a).
I think vaarali's explanation may have been valid. However, in the thread about Duryodhan that I linked to, in the earlier pages where we discussed the circumstances under which Krishna could support Arjun w/o violating any protocols, it was suggested that since Kunti was adopted out to Kuntibhoj (who along w/ his son Purujit fought & died on the Pandava side), she was not legally recognized as a part of the Yadava family, and so Vasudev & others were under no obligation to protect her or her children. If that were true, then Kunti's sons wouldn't be maternally Yadavas either - they'd be the descendants of Kuntibhoj from that side. Especially Karna, who was born in Kuntibhoj's kingdom (incidentally, any idea which kingdom Kuntibhoj ruled?)
Actually, in the Mahabharata, it's explicitly stated that Bheeshma approached Shalya or his father for Madri's hand for Pandu, and one reason for that has been the speculation that he was dissatisfied w/ Kunti's pre-marital birth to Karna, and therefore didn't want her to be the sole queen of Hastinapur. Beyond that, however, he decided to leave it up to her on whether to confess or not. Coming to think of it now, had she confessed, Duryodhan, Shakuni and even Dhritarashtra would have smeared both her and now all her sons - including Karna. Although that would have changed the equation against the Kauravas completely.
However, although I think that vaarali is legally correct, I think that her assumption and yours - that Karna would have been the emperor - is correct. In one case, Duryodhan, having befriended him, might have been okay w/ Karna, instead of Yudhisthir being the king, but on the other, he may well have used this revelation to try and disqualify the Pandavas on the basis of Kunti's character flaws. On the Yadava side, had Karna been accepted by Kunti on the day of the tournament, then I don't think there would have been any opposition - Krishna would have supported him, and so would Yudhisthir, and if Krishna supported him, he'd have automatically gotten the wholehearted support of Bhima & Arjun as well.
Also, did Pandu give up his rights to the throne for good, or did he take a break to do penance, and intended to return, except that he died b4 he could complete his austerities and return. If it was the former, I don't see what claims Yudhisthir or any of his sons would have had to the throne.
I too don't think that Vrishaketu was the eldest of his brothers & cousins. He was certainly the youngest of Karna's 5 or 7 sons, and younger than Abhimanyu, Iravana and Draupadi's sons. Question is whether he was younger than the potential survivors of the Pandava sons (assuming they survived) - Yaudheya, Sarvaga, Nirmitra & Suhotra. But even they weren't made successors to Yudhisthir. I'm assuming that Meghavarna, being a rakshasha, was ineligible to even live in Hastinapur, let alone rule.