Dear brother, wasalam and thanks for your appreciation.
12-15 days will be enough for your trip. I have recently done another 10 days trip to GB, started end of August and ended start of September 2018. You can visit shogran, naran, babusar, gilgit, naltar valley, hunza, (hopar valley) nagar, khunjerab pass. for naltar valley and probably hopar valley, you would require 4x4 jeep. for the rest of places, your can easily do in your own car.
1) I believe babusar will still be accessible onward 15th Oct 2018 but I am not 100% sure. Perhaps some one else can give your more precise info about babusar pass accessibility. weather in naran would be pleasant at mid-day and quite cold at night. Same goes for Hunza. Gilgit will have moderate weather i guess. Khunjerab will be very cold in mid october even at daytime due to cold breeze.
2) Yes you can go till khunjerab pass on your own car provided you have some experience driving in mountain areas as they have some steep ascends and descends, sharp turns, bad patches etc.
3) Best route is through babusar pass if it is open in october by the time of your visit.
4) In GB, places to see are Naltar Valley + Naltar Lake, Hopar Glacier in Nagar, Rama Valley + Rama Lake, Deosai (from gilgit side). These places require 4x4 jeep drive. In your own car, you will pass through Gulmit, Passu, Sost on your way to Khunjerab Pass.
In Gilgit, you can stay at PTDC, Riveria Hotel, Canopy Nexus etc. Most of hotels are on river view road. Or you can stay at Serena Gilgit as well though its expensive around Rs. 14000.00 per night including breakfast.
regards