First of all, the weather can be a bit unpredictable in October. It's starting to get colder now and most days are still sunny with a temperature of around 20 degrees Celsius, but sometimes it will shoot right up to high 20s/low 30s even in October, and sometimes it can rain a lot and go down to about 15. Anyway, the weather forecast shows good weather-sun every day with temperatures of 20-25 degrees, so you're in luck. Also, it shouldn't be too difficult to get around, especially if you have a GPS.
It will be hard to see everything Romania has to offer in just 5 days (in fact it's impossible), so I'd suggest coming back for a few more days next time you're in Europe. Your list looks good, but I'd also include Tulcea in the south-east so you can see the Danube Delta. It is about 281km/3.5 hours away from Bucharest but the problem is that it's out of the way for you, since you're going north-west and the Delta is east. You would need at least a whole day to drive there from Bucharest, stay for long enough to enjoy it and then drive back. I would therefore recommend leaving it for the next time you are in Europe, and we can focus on Bucharest and Transylvania now.
Bucharest is a beautiful city and it is said to resemble Paris (most of the buildings were actually designed by the same architect). I would recommend at least 2-3 days normally, but since you are so pressed for time, stay there for 1 day and move on. When you go up into Transylvania from Bucharest, I'd recommend taking one of two mountain roads: The Transalpina or the Transfagarasan. The Transfagarasan is a very dramatic road which is full of twists and turns and is said to be the best driving road in the world. It is also home to Balea Lake, which is a very beautiful glacial lake. The Transalpina, on the other hand, has very picturesque scenery (if you are going back to Bucharest afterwards to catch a flight, you could go on both). The Transfagarasan will end in Transylvania halfway between Brasov and Sibiu, whereas the Transalpina ends close to Hunedoara. The problem is that both of them leave out Sinaia, so if you really want to see Peles Castle, you should go to Brasov down Highway 1 instead. If you do so, you can see the Sphinx in the Bucegi Mountains on the way.
Whichever way you go, I strongly recommend seeing Sarmizegetusa Regia as well. It is the ruins of the old Dacian Capital and it lies between Hunedoara and Sibiu. Cluj is also worth visiting, but you don't have time. If you want, you can substitute Sighisoara for Cluj. Sighisoara is more like Sibiu and Brasov, whereas Cluj is the second biggest city in Romania. If you go to Cluj, you can see the Rachitele Waterfall and Tudra Gorges, both just outside Cluj. On the other hand, if you are going to Sighisoara, you can also make a quick stop in Bietran, which is between Sibiu and Sighisoara, and see the fortified church.
You should decide between the Transalpina, the Transfagarasan and the Sinaia/Sphinx road and whether you want to go to Cluj or Sighisoara based on what you feel you personally would enjoy more.
Other than that, you should spend a day in Bucharest, a day in Hunedoara and seeing Sarmizegetusa Regia, a day in Sibiu, a day in Brasov and seeing Bran Castle and a day seeing either Sigisoara and Bietran or Cluj, Tudra Gorges and the Rachitele Waterfall.
Next time you go to Romania, you must go to Tulcea, Constanta (if you're going during the Summer), Timisoara, Cluj or Sighisoara (depending on where you went now) and all around Moldova (stop in Iasi) to see the many monasteries there. You should also see the Rodna Mountains, spend a bit more time in Bucharest and go down another one of the roads which I mentioned. Enjoy your trip :)