Regarding yeast nutrient for cider making: I usually add about 200 grams of of plain light malt extract to provide nutrient. Mainly because I rarely see commercial nutrient for sale. Anyway I believe the malt improves the flavour profile.
Day 13 seems quick, mine is currently on day 23 and the gravity is still dropping (1.010 on day 21). I am fermenting at 12°C which might slow things down.