Don't cut them off ! Otherwise you'll miss out on the intermediate and upper pitchers, which can usually be different in shape and colour, depending on the species though. Not all nepenthes put out different uppers. You're doing the right thing by supporting the vine, just use anything suitable to help support it, bamboo sticks, trellis, ties etc. Can you post a photo of the plant in question ?
As for your new plant, it's still a bit early to tell, but it looks to me like it's very likely to be x ventrata, a hybrid between alata and ventricosa. This is a very common 'starter nepenthes' due to it's tolerance to less than perfect conditions and it's easyness as a windowsill plant.
Happy growing !