If you want to run Tomb Raider and Batman Arkham at a respectable frame rate, you aren't going to do it at $750 or under. It just isn't possible. Especially with a 2-in-1 that has a touch screen.
From everything you listed that you will be doing, those 2 games will require more processing power than all the rest combined.
If this is for school and classes, you really need something lightweight and with a long battery life as well as good processing power, but you won't find it anywhere near that price, I'm afraid.
I'll see if I can find some ideas later and post back with some suggestions, but for right now, I'm at a loss.
I have seen the Lenovo Flex 3 14" 2-in-1 for $799 with a 2.4GHz i7-5500U dual core, 8GB RAM, and a 1TB + 8GB SHHD (Hybrid SSD and HDD). You might start looking in that direction. Perhaps someone else has seen a laptop that will come closer.
Honestly, that laptop has the integrated Intel HD 5500 graphics, which will be fine for almost everything you'd need to do except gaming.
You would more than likely spend double your budget on a laptop that can handle those games, and they won't be lightweight nor would they have long lasting batteries.