I definitely agree that her personality insofar as she cries like hashi did is similar, nice find. But everything else can be explained off, and no im not saying that because i dislike sakura.
For example, Sakura was always very intelligent, she was able to break free of genjutsu in the chuunin exam arc. So it comes as no surprise that she knew that the first hokage used mokuton, that doesnt establish any connection of her to the first. Using your logic i can argue that jiraiya is related to the rikudo sennin or will get powers like the so6p because he knew the sage had the rinnegan when he saw nagatos eyes.
Also about them having the same symbol i wouldnt really go there. I can show you 10 instances of the gedo mazo having the uzumaki clan symbol marked on it but as we knowthat to statue has nothing to do with that clan as i myself falsely believed when i made a theory about it long ago.
I wouldnt doubt that tsunade having a mark on her forehead proves her connection to hashirama since its canon manga fact that shes his granddaughter.I dont see how this connects to sakura though.
About yamato teaming up, well she didnt get along with sai if you recall so that could also be the reason.
Her figuring out zetsu can also be attributed to her intelligence and superior medical training under tsunade.
Umm, it was stated in the image you yourself showed that she knew neji was a zetsu because of the question she asked about ton ton, not because shes connected in anyway to the 1st.
About her getting stronger with senju like powers, it cant hurt cuz the juubi is screwing them over lol. But personally i think kishi had manyy chances to make her more strong and develop her character, im not sure if he will now but lets see what happens.