Yi Xing Long = Omega Shenron/the final dragon
Si Xing Long = The fire dragon that becomes an ally eventually
I'm a big GT fan but it always bothered me how powerful YXL and the rest of the Evil Dragons were. The Dragon Balls can't even kill Nappa, but the negative energy from wishes being made is somehow enough to generate a being that is almost SSj4 fusion tier?
In light of Battle of Gods there may finally be an odd way to explain them if you can accept that GT and BoG happened in the same timeline.
After Beerus and Son Goku fight in Battle of Gods, Beerus mentions that there is a multiverse and that Beerus is only the God of Destruction in the seventh universe (the only canon one.) This makes it pretty clear that there are up to 11 other Gods of Destruction that we are not aware of.
In GT when Yi Xing Long kills and absorbs Si Xing Long again he remarks that he is a "god of destruction."
While I realise GT was made decades before BoG and Toei just wrote that as a grandiose statement and not as a reference to something that Toriyama came up with much later on, his power would make a Lot more sense if he was indeed a literal God of Destruction who came to Earth from a parallel universe.
How did a God of Destruction from a different universe come to Earth through the Dragon Balls and why were he and his minions so closely associated with the Dragon Balls? Furthermore why does he even Have so many underlings?
Regarding the first issue, remember the last wish made on the Earth's Dragon Balls was made in the Boo arc. The Evil Dragons arc took place 16 years after that. The wish that turned Son Goku young was made on Piccolo Daimao's dragon balls, as was Baby's wish to recreate the Tsufuru-jin planet. The wish to restore Earth was made on the Namekseijin DBs.
If the Dragon Balls only became corrupt then, then it was not just the wishes on the DBs that were at fault here.
The DBs only became corrupted after Dr. Mu and Dr. Gero screwed with the fabric of the universe by attaching Hell to the living world and invaded the living world with dead villains.
Between powerful magical objects that can create entire planets, resurrect people, etc being corroded with negative energy and the dimensional crisis caused by the living world and the world of the dead becoming one, it's not a stretch that a powerful enough being like YXL could use this to invade this universe.
In the original DB, people on par with Evil Boo/SSj3 Gotenks or higher could scream holes into other dimensions, and in Anime filler Gohan-Boo was able to scream the entire universe apart and threaten to collapse it.
An incomprehensibly more powerful being like YXL being able to travel the multiverse by taking advantage of a situation like this is not unthinkable.
It's also worth noting this interesting tidbit from the GT Perfect Files translations on this website:
It's not flat-out saying they come from a different universe, but that's likely because the multiverse was yet to be introduced and this was the closest they could go without going a bit crazy. We have no idea how exactly the multiverse works and where or how the universes exist. The simplest idea would be that they're just giant spheres or whatever situated next to each other, so from that interpretation it's possible the galaxy could be in a parallel universe and still be "neighbouring" the galaxy the Earth exists in...For a long time, the original Dragon Balls were used no more than once in a hundred years. But, with the advent of Gokuu and friends, Shenlon has been summoned countless times over the last 50 years. This has caused a build up of negative energy within the Dragon Balls. Pretty soon, the Balls will crack open and release that energy. The negative energy will then attract 7 evil Dragons from a neighboring galaxy. Each Dragon has a DB somewhere within its body, and is named after their corresponding Dragon Ball's true name. That is, the Dragon with the #2 Ball is called "Ryan-Shinlon."
As for why YXL is so closely tied to the DBs, it's possible that his universe also has (a very different and much more powerful set of) Dragon Balls and his origins are tied to them, or that the Dragon Balls are simply his link back to his world.
One thing that always puzzled me was how Si Xing Long could die and then simply return from his DB. Also the issue of how Piccolo could handle the Evil Dragons if they went to Hell.
In light of this it's possible that none of them died, they were simply sent back to their own world and could continue coming back all they wanted (like the villains that escaped Hell) if it weren't for Shen Long taking the Dragon Balls with him for 100 years, possibly to purify them of whatever YXL did to them to make them usable as a gateway.
As for who his minions are. I imagine they're simply parts of him in the way that Cyborg 17 and 18 are part of Perfect Cell and he purposefully unleashed them first so he could spy and see what he's dealing with in this new universe.
He Did say he was a God of Destruction when he had re-absorbed all the DBs so for the purposes of this theory at least, his Super form is his true form and the only reason he wasn't in it to begin with is because of the whole preferring to send out a bunch of extremely diverse minions to get a feel for what he'll be up against.
Battle power wise I feel Super Yi Xing Long is a good place to put Beerus if attempting to make a list that includes both BoG and GT. It's arguably implied that Beerus is on a level unimaginable to even someone who has the memories of the almighty Vegetto and most likely knows how powerful SSj3 Vegetto would be. Yi Xing Long should also be vastly superior to any form of Z Vegetto IMO going by Super Vegeta-Baby > SSj Vegetto > Vegeta-Baby.
EDIT: After some more thinking, that has to be it. Yi Xing Long being from Universe 7 makes no sense whatsoever. He made it clear that his goal is to obliterate the entire universe and Elder Kaioshin's vision made it clear he would soon do just that.
Unless Yi Xing Long was just peacefully lounging on a couch somewhere deep in another galaxy in Universe 7 and suddenly flew into a universe-destroying rage over corrupted Dragon Balls, he had to have come from another universe altogether IMO.
It would explain Yi Xing Long's motives a bit more. If he's a Hakaishin then he's a being whose only purpose is to destroy things. However, Whis stopping Beerus during his solar system busting rampage at the end of BoG implies there's checks in place to stop a God of Destruction from systematically destroying the whole universe out of boredom.
Perhaps he relocated to Universe 7 since he was not the Hakaishin there and thus was a "neutral" being like Freeza and could run wild and kill/destroy as much as he wanted without consequence at long last.
Son Goku wanted to stay dead after the Cell arc due to believing himself to be a magnet for evil. Him leaving for 100 years with the Dragon Balls is a nice callback to that since now even the corrupted DBs have become a magnet for unstoppable evil beings from other worlds.