Yes, you can absolutely split your site into sections like you describe and use simple include() statements if PHP is available or SSI directives to call them. Before the rise of CMS platforms, this is how many people created sites with consistent and easy to modify sections.
Building your pages that way has no positive or negative effect on SEO. So long as you include all of the necessary elements for SEO (title tags, meta description, rel author, etc) then your site will do as well as it did before.
What CMS's like WordPress do for you is force you into "better" (not best) practices for SEO by doing things like requiring titles for all content and automatically generating a menu system that a spider can crawl. Via the use of extensions/modules/plugins you can add more advanced SEO techniques to the site but you can also do it by hand on a non-CMS site. Ultimately working with a CMS makes things easier for some people as it automates a lot of aspects of site-building. But that same automation has drawbacks and some people prefer to hand-code everything and retain pinpoint control over all aspects of the page. Ultimately you will decide to do what is best for your development style and site needs.