Isnt it "Peter DOES not see the woman?"
That's also a possible translation, and a more literal one.
Why "Péter doesn't see the lady" was not accepted? ?
"Lady" translates better as hölgy, although there is some overlap.
Literally, "Peter isn't seeing the woman". Duolingo sometimes uses a normal English expression (as in the answer) and sometimes it only accepts a literal translation. Never sure how to second-guess it!
"to be seeing" has fairly different connotations, you could take it as a special phrase
Yes this should be does not see
Does anyone know why it's látja and not lát or látsz?
"Látja" is 3rd person definite (because it's "THE woman"); "lát" is 3rd person indefinite, "látsz" is 2nd person indefinite. (See "Tips & Notes.")
"doesn't see"="nem látja", "cannot see"="nem láthatja"