2 big plotholes in death note story (Spoilers)

[Okami]

I just finished watching death note and I found two big holes in death note story and since it it such popular anime I thought of sharing them here.

-First one is about Shidoh and his death note, after thinking a little bit I figured out that his death note couldn't be one that ended up in Mellos hands but is eather one that was with Misa or the one that was with Ryuk. How did I get to this conclusion? Pretty simple, you just need to follow death notes swamping carefully. In beginning there was 4 death notes in the human world Ryuks note that was with him, Shidoh note that Ryuk stole from him and dropped it in the human world that Light found, Rems note that was with her and note from dead Shinigami that died protecting Misa that Rem picked up and give to Misa. Now the first time Light started swamping death notes was when he loses his memories so that he could persuade L that he isn't Kira. That time he buried his note (Shidoh note) and told Ryuk to take away his ownership of it once he told him to and he give away Misas note back to Rem and told her to give it's ownership to someone who will continue killings. So the one note that Rem gives to Higuchi is Misas note (The one from dead Shinigami). After L and Light catch Higuchi it is that one note that ended in hand of the L's department and Light has taken the note from Rem after she died killing L and saveing Misas life that way then after that Light has told Misa to dig out the note that was originally his (Shidoh note). So at this point it is like this Light = Rems note, L's department = Misas note (the one from dead Shinigami), Misa = note that light originally picked up (Shidoh note) and Ryuk = his own note. So the note that Mello has taken from the L's department is the one from dead Shinigami and Shidoh note is with Misa (Even if we say that Ryuk has originally dropped his own note and kept Shidoh note with him it wouldn't change anything as than Ryuk would have Shidoh note and Misa would have Ryuks) so Shidoh note can't be the one that ended up with Mello. Some might say that this isn't such big hole in the story bad may I remind you that without this Shidoh would never talk to Mello and Mello and Nate wouldn't find out about fake rule in it and so nobody would started suspecting Light to be Kira again.

-The second Hole is pretty simple, Light newer found out Mellos real name but somehow Takada has known his name when he kidnapped her. She could't had the eyes as she never had ownership of any death note and was only using pages from it so that is not an option.

Also in the end Mikami could simply use the death note that was with him to confirm that it is real instead of doing tests on it. But if I start talking about this I will find another 50 flows within the Nate arc and the ending but they are not really holes in the story just flows so I won't start talking about it now.