Yes a video background does effect your SEO see https://support.google.com/adsense/answer/6196932?hl=en and http://searchengineland.com/mobilegeddon-beginning-not-ending-220512
The slower your site loads the farther Google pushes you down the SERPs. The biggest concern for Google is mobile devices.
So what should you do instead of adding a video background? Research Sales Copywriting, customers care about your website solving their problem more than anything else. It's hard for a video that's a background to solve any problem unless that customer is looking for flash.
Maybe, but the main goal of your site should be to create something users love. YouYour website could look terrible and very plain and outdated however if userusers find it valuable you'll get backlinks and organic traffic. See http://craigslist.org for a shining example.