Definitely has something to do with the manipulated Lineage Factor in my book. We know the RS is tied to his DNA and that's why only he can use it, as noted by Luffy and Chopper. So when he puts the RS on, it's connected to his DNA and that's probably how it affects his genes. I'm guessing the third use of the RS will cause the transformation to be complete and give him eveything he was supposed to be born with : exoskeleton, accelerated healing, super strength, enhanced speed and maybe even black hair.
As for the whole fire thing, we know it's not something natural at the very least, as established by Queen's statement. Oda is telling us no human can just burst into flames for no particular reason, period. There are a lot of possibilites but I wouldn't dismiss the possibility of a connection to Lunarian people like some people did, I think it would actually make a lot of sense.
I mean sure, Queen's sentence sounds like he was just introducing the name of King's race, but I find King's reaction to be really intriguing. If he was only reacting to Queen referring to his race, why did it catch his attention to the point where even Zoro had to make him focus on their duel ? To me, it really seemed like King was suspiciously intrigued by Sanji's ability, rather than just reacting to Queen mentioning his race. Besides, there was no need to use Sanji's DJ as an excuse to reveal King's race, Marco or anybody else could have brought that up. I was really under the impression Oda was just creating a connection to justify something that will be revealed later. And that thing would be Sanji having Lunarian DNA. It would give a whole new meaning to King's reaction, who might have realized he's not truly the last individual of his extinguished race (although I'm not sure whether it's been eastablished King is officially the only survivor).
Yes, it could be argued that there would be no reason for Judge to only give Lunarian DNA to Sanji, but let's not forget the special power he was supposed to be born with was invisibilty, which is a passive ability. The other siblings have a destructive power they can use to destroy everyone and everything around them. So it can be speculated Lunarian DNA was only given to Sanji so he could have an active power to use in combat. Maybe Judge didn't even have enough DNA for all of them and had to use it for only one child.
Another possbility is of course that Sora is a Lunarian herself but that would be harder to justify. She had no Lunarian attribute as far as we know, no flames and no wings. And why would Sanji be the only one to receive her Lunarian genes ? Makes no sense. Although her name meaning "sky" in japanese is an interesting fact, since the term "Lunarian" is reference to the Moon.
Curious about it being “since the second time.” That would be… two odd weeks ago at the bathhouse that Sanji has been feeling odd?
The difference between the second time and the first time with Page One, that I can remember is that Sanji had a nosebleed. Could his blood have activated something in the RS that in turn has effected his body?
I think it's more likely that Sanji didn't take off his RS after his fight with Page One, and thus his use of the RS in the bathhouse still counts as first time use. The second time would be when he saved Momo and faced King.
Speaking of which, we could speculate that Sanji surviving King's attack had something to do with his super genes starting to kick in. I mean he didn't expect to survive the attack despite being already aware of the durability boost offered by the RS. If that's when he started feeling weird then maybe his exoskeleton partially awakened.