if you havent noticed the cursed seal increases Sasukes power tonnes, level 2 will increase it far more then Sasuke would ever achieve by himself.

Itachi left the leaf because he could no longer become stronger there seeing as he already killed his clan, the main reason that Itachi left Sasuke alive that we know is to beat him.
Sasuke is there to one day beat Itachi, thats what Itachi wishes him to do but obviously only when he thinks hes a worthy challenge.

Kakashi WAS keeping up with Itachi before the mange sharigan, he had to save Asuma and Kunerai most of the time thats why Itachi was winning. I suggest you read again you'll notice that Kakashi could have still fought for a fair while with Itachi if he didnt try to take the mange sharigan on head on.

Kakashi thought he could counter the mange sharigan with his eye thats the main reason why he was beaten so badly. Kakashi has the bad habit of making mistakes the first time around as seen vs Zabuza. If you ask me Itachi is over rated just because he beat Kakashi with the mange sharigan, i mean Zabuza came close to beating Kakashi too many times.

this is NO way shows that Itachi can take on someone of Jiyairas level, Kakashi and Jiyaira are at a huge gap in terms of power.
Even if Itachi can beat Kakashi it doesnt bring him close to Jiyaira by any means