Yes of the 3 the Asus Q550LF-BB17T07 would last longest. It has a better CPU and a better GPU, which is really what the power comes from. It will do best in games and for general use like web browsing, writing papers, office work, etc. I am not sure if you can buy new ones still or not though. Maybe some where, but it would probably be over $1000. The GPU being the important part would be twice the speed of the HP laptop I would say.
If you go with one of the others, the other Asus system will be a little faster when it comes to having a lot of we pages open, and office work, or compressing or uncompromising files. The HP, a little slower than this still, but it will be considerably faster in games. The Asus Q550LF-BB17T07 will give the best of both, with faster CPU performance out of the three, and it has about 35% faster GPU than the HP, making it considerably faster too.
If you don't like buying used, make a new thread asking which laptop you should buy for the price you can go up to, what you want to do with it as specific as possible, and what features are really key that you want, for instance touch screen seems to be something you want.
If you decide to stay between the two New systems discussed here, I would honestly say go with the HP. HP isn't as good as Asus in terms of quality, which might end bad for you, but all the ones I have seen, while they have had some issues, are still running and they are more than 5 years old. My grandmothers took getting tossed around the room and its still kicking too. I think people just over do thinking they are terrible. I wouldn't personally want them, but if it had better parts for the money I would still buy it.
As for why I would choose HP, while the CPU is faster in the ASUS, you won't notice it as much. Even if you just want to play games on medium settings, you would notice the better GPU a lot more than the better CPU. Think about it, both run all your basic Office programs and internet browsers, video players and such without issue. It might take them a second longer on the HP, but its just a second. While in games the difference is going from low to medium settings with several games.
As for buying Refurbished, I understand why you may not want to, but its not so bad as you might think. I bought a refurbished Samsung laptop last October for $350. I put an extra $200 into upgrading the CPU, HDD, and RAM. I overclocked the GPU, and now sitting at $550 I have a laptop which is more powerful than all of these. The better Asus laptop has a slightly better GPU than mine, but its only about 5% faster when it only has 64-bit GPU. So buying a used or refurbished system can sometimes work out really well.