How did Snape allow his old copy of "Advanced Potion-Making" to be
used by Harry?
He wasn't aware of it until the 'Draco - Sectumsempra' incident.
In the movie: And as soon as Snape realises that Harry might be in possession of his potions book, he questions and tries to get the book (which Harry hides in the room of requirement for fear of Snape confiscating it)
In the book: Harry switches his book with Ron's and ends up showing Ron's copy to Snape claiming it as his book.
Remember that Snape did not teach Harry potions in year six, it was Slughorn. Had Snape been the potions teacher, he would have immediately caught on that Harry was using his old book.
And when Slughorn mentioned that Harry had a flair for potions, Snape was sceptical but never doubtful. He probably didn't even remember his book.
Was this by the instruction of Dumbledore in order
to let Harry get closer to Professor Slughorn and therefore find out
more about the Horcruxes?
Harry didn't need to even try to get into Slughorn's good books. Slughorn was anyway going to pull Harry into his 'Slughorn Club' as Harry would be his most famous student and club member. The 'potion flair' that Harry had was just an added advantage.
Coming to if this book was planned to be with Harry - No, its a mere coincidence
The potions master for that year was Slughorn, and since Harry and Ron were already sure they would not be in Snape's potions class, they never purchased potions books for year 6.
When they realise Slughorn is the potions master that year and they could take up potion classes, they were asked to take spare books from the shelf by Slughorn. When they find two books - one new and one battered, both of them quabble over the new book which Ron wins. Harry just settled for the battered one - which is purely coincidence.
Snape actually misplace the potions book?
I don't really know if this is explicitly mentioned in the movie/book but it could just be that he left it there long ago and didn't really remember.