If you are looking at spending $500 on a new laptop, then you are basically limited to integrated graphics. The following $510 ASUS VivoBook F510UA is generally the best laptop you can that is witihn your price range. It has a quad core i5-8250u CPU, 1080p screen, Intel UHD 620 graphics core, 8GB of RAM and a 1TB hard drive.
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/www.amazon.com/VivoBook-i5-8250U-NanoEdge-Fingerprint-F510UA-AH51/dp/B0762S8PYM
Steam sells a lot of games so the term "Steam games" is too generic. Simple games like Shovel Knight, The Binding of Issac, and Terraria should not be problem. Half Life 1 & 2 and Portal 1 & 2 should not be a problem at all. League of Legends is yet another game you can play; though it is not on Steam. You can even play Fallout...