I've been using a Gateway laptop with an i5-3337u/HD4000 setup with 8GB of RAM, and while you're not going to be running new stuff real well, I've had no issue with games like Skyrim or ME3 on low settings and slightly lower than native resolution. TF2 runs great at native res, as do the few indie games I've tried (mostly, Rogue Legacy). I managed games like Oblivion, Modern Warfare 2, and a lot of classic D&D games, KoToR, etc. even on a pre-HD Intel chipset; I see no reason they'd be any worse now.
Most of the stuff you want to play will probably work fine, although with HDMI-out you might want to keep things at 720p or so. I haven't found Steam's big picture mode to be really well optimized either, but I'm also limited to 128MB of dedicated RAM for the HD4000 the way Gateway's locked my BIOS.
If there's anything specific you want me to test, let me know and I'll try to try it.