Picking a chassis size and style you like, plus functionality you want or need, i.e. touch vs non., glossy or matte screen, 1080p or 4k, convertible or classic laptop, storage needed,etc. matter as much or more as i5 vs i7. Unfortunately, some manufacturers tie the RAM config of the laptop to which processor is installed, so you may have to go i7 if you want max RAM.
Gaming is one of the most challenging asks for a thin and light laptop at a reasonable price. For the most part you're looking at a 15" chassis and fairly thick for systems with a solid GPU. Razer Blade 15" is thin, but still 4.5lbs and pretty pricey
https://www.razer.com/gaming-laptops/razer-blade
If you're leaning towards something on the lighter side, and willing to give up gaming on the smaller laptop, consider the Lenovo X1 Carbon, Surface Pro and Surface laptop. I've used a Thinkpad X1 Carbon 3rd gen and a Surface Pro 4 in the past year at work. The Surface was nice for screen quality and low bulk, but I preferred the traditional laptop hinge of the Carbon for the ways I most frequently use laptops. X1 Carbons and Surfaces are not going to be among the less expensive options, but fit and finish are generally pretty nice.