Regular Eirika's weapon evolution

I don't see why not. Her Brother got his legendary weapon as an upgrade too. So why not her

Its iffy
Hector got LAHector but not Lectors axe
Brave Lyn didn't get Swift Mulagnir
Ephraim got LEphraims lance
None of the Tikis got Divine mist (or summer breath)
LRyoma came with Rajinto
LGrima came with Expiration
Like got Ragnell from the get go, so realistically I could see it, since it is more or less a sidevolution (Eldigan and Ares, Seliph and Sigurd, Hector and LA Hector, and Ephraim aaaand LEphraim) I think it'd be interesting to see an offensive Eirika (especially considering her naturally high base speed, getting added defense and CDR could be nuts) but traditionally Support Eirka and her prf refine will stay the meta

I don't think so. They stopped giving weapons of Legendary Heroes to older versions of said heroes.

Ephraim didn't get it immediately after. Ephraim's weapon refine for Siegmund itself was pretty bad too, so perhaps they felt they had to give him flame siegmund which gives him more attack but splits him off from the team

Either way, regular Sieglinde has a higher potential ceiling and is easier to use on an infantry unit. Trying to be outnumbered at the range of two spaces requires running pretty far away which is easier on a cavalry unit.