The good the bad and the ugly.
The good: I like the core of ideas you've created. You have an interesting background and some cool concepts and a great way to tie in the world you've created.
The bad: Though you decided to use a missing primarch, you forgot one detail. The Tyranids didn't invade the Milky Way until long after the primarchs had moved on into history.
The ugly: Certain words and ideas seem a little "clunky". Plasma Eagles kind of stumbles off the tongue rather than flows.
A few changes (if I may be so bold):
Keep your primarch idea. His name is a mouthful but that just comes with the primarch territory. Use the story of being raised by eagles as a myth rather than fact.
The idea of a demi-god having his breakfest puked from a giant eagles mouth is a bit silly. You also didn't really descibe the primarch. What is his unique mutation?
Feathers instead of hair? Talons instead of fingers? Perhaps he is a master of reading body language and thus can communicate with any animal.
I see this primarch with a falcon on his wrist and the wind in his hair.
You also said that he was making lighting claws before discovery by the imperium? That's fine (Lion
El'Johnson had power armour on his homeworld) but you might be able to switch it up a bit from there.
If his planet of Boeing (all I can think of is big planes) had
STC info maybe they had other artifacts too. Jump Packs and Lightning claws would fit your hawk theme just fine. That would also allow the primarch
to have access to flight and claws before his discovery of the imperium.
About the Lions, did you include this part of the story just to have a "griffon theme"? I'd drop it. You already have the core creature type your chapter/legion emulates.
The planet itself could use a bit of alteration. With plamsa lakes/rain/clouds, it seems like the planet is more of a cooling star or massive dust cload forming a planetoid.
Also, are the eagles of the world somehow evolved to handle such a toxic atmosphere? Perhaps they have electro-magnetic feathers that sheild their bodies.
Maybe these feathers could be used as power blades and such. Plasma could also be the core weapon of your chapter/legion.
In closing I'd say you have the start of a really cool idea but you just need to iron out some of the details. Form a personality for the primarch and his legion.
Determine what form of culture the world had before and after the coming of the primarch.
Change the name from Plasma Eagles to something a little more original (Flame Falcons, Blood Ravens, Doom Eagles, Raven Guard and Howling Griffons are all bird related).
Maybe something like the Storm Falcons, Condors you get the idea.
Again, these are just some quick thoughts. I'm not trying to tell you how to do your thing.
PS If you want, I'm currently drawing a bunch of space marine pictures for people here on dakka and am posting them in the
40k General Discussion forum.
Once you decide what the primarch looks and acts like, let me know and I can render him in my loving remembrancers hand.