I remember seeing a special about the POTA movies some years ago, and I'm pretty sure they said that Battle was the only movie written with a sequel in mind. Certainly the ending of Escape was meant to be ominous, to show that the seeds of the apocalyptic future had been sown, but as I said, I think the idea there was that it would be many generations down the road, not just a couple of decades as the next movie had it.
That's correct, from the film itself. Cornelius's testimony in Escape
puts the ape uprising generations after his presence on Earth.
They replay some of that testimony in Battle
, but snip the time indications out of it.