Is it ok to chop index.php and include files?

Yes, you can separate your code and markup into multiple files. Be aware of scope issues when you use include, but done carefully it should be no problem. Since you are in WordPress and building a theme, I would suggest get_template_part over a raw include.

In many ways, “chopping” up your theme files is what get_template_part is for– well, at least it is very convenient for that. It lets you separate components of the page and include them in a way that makes easy to extend via child themes.