Lutecia's transformation into a Genki Girl makes sense when you consider that among the Numbers Cyborgs, she spends the most time with Sein and Wendi, and this continues when she's in the rehabilitation center. Presumably, their cheery personalities gradually rubbed off on her, since she seems to like that part of them. That and she has her mom back.
Corona mentioned that when the Yagami family participated in their training match/mock battle, they completely dominated the field. Another case of Informed Ability right? That is until you remember what kind of mage Hayate is, and the kind of magic she's capable of. And with the best possible protectors, all she has to do is wait until her spell is complete, tell her knights to get the hell out of the blast zone (or have Shamal teleport them out), and nuke the field, effectively eliminating all opposition in one swipe.
Before their match, Sieglinde Jeremiah stated she had a good idea of Einhart's fighting style despite the matches ending in only one hit. She most likely got most of the things she needed from Einhart's match with Corona. The match itself took two rounds, which allowed Sieg to notice most of Einhart's strengths and possibly her weaknesses.
Vivio is the clone of Olivie, who was recognized as the strongest warrior in the ancient Belkan era before her death. But while she's no pushover herself, Vivio still isn't on Olivie's level. Why? Partly because of a different upbringing but mostly because while Olivie had a Relic Core (inherited from her mother) in her body her whole life, Vivio's was destroyed (or maybe just some of it) by Nanoha in the Cradle at the end of StrikerS.
It also makes sense that Vivio did not develop a body suitable for combat because Olivie's body was also physically weak and had poor magical reserves, with only the relic inside her giving it unparalleled magical reserves that allow Olivie to reinforce her body. What Vivio is as a fighter today is what Olivie would've been as a fighter if she did not have a relic.
As ViVid Strike! shows, Miura is still stronger than Vivio as evidenced by the former being two spots higher in the U-15 world rankings than the latter. This can make Miura come across as a Creator's Pet to some, but in the context of magical stats, Miura has the two things that Vivio sorely lacks: strength and defense. Her speed is slightly lacking compared to Vivio's but she hits way harder and endurance is considerably higher, which proves to be the reason why she is able to beat Vivio in their first official match. Miura is essentially the more balanced Lightning Bruiser compared to Vivio's Fragile Speedster, and if Einhart's matches against Vivio are anything to consider, Vivio is outmatched unless she can read all of Miura's attacks flawlessly or she activates Sacred Defender and becomes a full on Glass Cannon.
Rio has several Chinese elements and Micaiah has Japanese, but isn't Earth an insignificant little blue planet? Well, Subaru stated way back in StrikerS that her father's ancestors were from Unadministrated planet #97, so it won't be too surprising if others of Japanese or Chinese descent were living in modern Mid-Childa.
Sieglinde's Kansai-ben dialect. She isn't from Earth, but Mid-Childa, but why does she speak Kansai-ben? Do people on Mid-Childa actually speak Japanese? The main language seems to be English as seen on several on-screen textes, and the Japanese words on Harry's coat are referred as a "foreign language".
Presumably it's a Translation Convention. She's speaking with the Mid-Childan equivalent of a Kansai accent and it gets translated as Kansai-ben (kind of how Kansai-ben is often turn into a Brooklyn accent in dubs). Also, this user thinks it's probably unlikely that Mid-Childa actually uses English as their main language (that being Earth creation) and again it's a translation.
Vivio's lack of strength compared to her original Olivie may not be because she lacks a relic like Olivie, but because her body hasn't properly adapted to it yet. In StrikerS the Relic is only shown cracking, not breaking, so not all of it may have left her and the rest may have become dormant within her. Sacred Heart likely gives her adult mode by turning it back on in her terms compared to the adult mode Jail forced on her, which likely forcefully drew it's full power into her body and giving her no control of it. Vivio will likely need about the same time as Olivie did before the Relic will properly adapt to her physiology.