I see a mouse here should also be accepted. TU means also here.
"I see the mouse here" should (I think) also be a valid answer—there's no demonstrative pronoun, so it should be able to take either article.
Disclaimer: I am learning, too, so this is a GUESS.
I think your suggested answer would require "tu" twice -- one for "tu" as in "the" and the other for "tu" as in "here."
What about I see the mouse? Thanks
That is also accepted.
When is it "to" "ta" "tu"?... I don't understand.. Is it "the"?
Often it stays for the or that. But sometimes we do not need any demonstrative pronoun when it is clear from context that we are speaking about one specific object.