Bounce rate might reduce if your site is visually attractive... people might be more likely to share your site with others too. But would it directly impact SEO? usually not unless it makes the page slower / faster.
Its possible that it may change. It depends on how your website is designed, does it follow the standards given by http://www.w3.org/? Is the web source easier to read compared to the old one? Change is normally good, search engines love fresh content.
if you currently have good ranks be sure to forward any 404's you create by changing page names to the new ones, or you'll mess up any SEO work you've already done
Changing the design actually help to make the visitors stay longer at the site and maybe browse to other page. In terms of seo, that would reduce bounce rate.
The structure of your website coding or programming has everything to do with your SEO. Search engines have certain things they look for in the html of websites.
I think, if you change content and names of content like alt tag etc, it effect on SEO. But if you change only design then it attract user as result user will increase or decrease according to design whether they like or not. But one thing, it should be SEO friendly.