Did Kyouko fight Sayaka in the new universe?
There are, definitionally, no witches in the post-Madoka universe. As such, it is not possible that Kyouko fought Sayaka's witch.
Besides, Mami says:

行ってしまったわ。円環の理に導かれて。
ミキさん、さっきのあの一撃にすべての力を使ってしまったのね。
She's gone. Taken away by the Law of Cycles.
Miki used up all her energy in that last attack, didn't she?

And furthermore, we see that they have just set a wraith/demon/魔獣 on fire. It is very clear from this that Sayaka just disappeared after running out of energy in a fight with a wraith/demon/魔獣.