This is mainly theory but i notice theres something that ties these all together and i just have some theories about all of it.
Dragon Lachrima we know is a lachrima that gives someone the ability to use Dragon Magic people who have this are Laxus, Cobra, Sting, & Rogue
3rd Generation Dragon Slayers are DS who were taught ds magic by a dragon and also have dragon lachrima inside of them.
The 3rd gen DS we know about lachrima appears to be the same type of DS magic as what they were taught
They both killed there Dragon teacher.
We know that dragon slaying magic evolves through killing dragons and "Bathing in the dragon's blood" is what was said but what if they don't mean literally the blood, what if Dragon Lachrima comes from a slain dragon similar to how when they killed Exceeds and lachrima was left behind. Seeing as they are both magical creatures it makes sense to me.
So If we were to assume that Rogue and Sting were originally Generation 1 DS Then as we saw killed their dragons and took the lachrima from them this would explain why they didn't have a seperate element lachrima and where they got the same element lachrima.
Which enables them to use Dragon Force
Then it opens up Double Element DS theory
So what if When natsu received all that magic power from laxus a 2nd Gen DS (Lachrima only) It basically left behind a small piece of Lightning dragon Lachrima, Not enough to use Dragon force but to access the double element attacks we've seen.
We know 3 double element Dragon slayer occurances (not counting other non DS magics natsus absorbed obviously)
Natsu with Laxus's Lightning
Future Rogue with Stings Holy
Gajeel with Rogue's Shadows
The thing in common from all of these is that the one whose power is being gained, has lachrima inside of them.
Few things all this info could mean
Gajeel may have some of Rogues shadow lachrima within him and be the 2nd double elemented DS
Dragons leave Lachrima upon death
Lachrima of your same element allows you to open dragon force at will.
Natsu may finally meet igneel only to see him die and get his lachrima then Dforce up, but lets hope not. I was hoping he would've got lachrima from Atlas flames leftover flame.
Laxus was using lightning magic before he became a gen 2 dragon slayer, so I always assumed that dragon lachrima didn't have a particular element, and just enhanced the properties of the mage, ex)lightning user becomes lightning dragon.
i always thought something similar to that... the fact that all dual element dragons slayers get their second element from a DS with a lachrima inside gave me the idea that ds can only absorb and gain control of a second element, if and only when it comes from a lachrima (a dragon lachrima), not the mage itself, this being the reason why natsu can still control laxus lighting(who as a lachima inside) while not capable of continue using the magic from another mage, like totomaru (whose fire magic should be more compatible with a flame DS)...
and i also have the idea that something similar happened when natsu got to use flame god slayer magic... after all, it was only stated that Zancrow obviously didn't learn it from a god, nor Chelia (who learned from a book), so probably a requirement for it, is to have a lachrima inside the body (we don't have enough information on chelia, zancrow, nor orga to confirm or denied whether or not they have a lachrima inside), which enabled natsu to gain momentary control over it, being this somehow similar to what happened with ethereon. yet because of the magic coming from a different kind of lachrima, or probably the difference in the very basics of the magic (DS magic- GS magic, DS-etherion highly concentrated magic), didn't allowed him to continue using it.
i do like the theory of of the dragon lachrima coming from a slain dragon, i never thought about it, but it could explain why they are so rare. but i don't think the DS get to form a lachrima inside their bodies, they just get accustomed the the new kind of magic, and get to generate it... that or they have it somewhere inside their stomach, (next to their primary element).. after all the whole body of a dragons slayer seem a bit like a lachrima (generating, storing, multiplying, adsorbing and overall controlling a specific magic). but who knows maybe you're right on that one.