When a website is remodeled, are the URLs of the posts and pages kept? Is the positioning maintained?

In most cases, you would want to keep the URLs of the posts and pages the same. If the pages had good organic rankings, maintaining the URLs is essential to maintaining those rankings. Additionally, if URLs are changed, and proper redirects are not in place, users will experience errors, and your site risks losing rank. Whether or not you change positioning is entirely up to you. With every website update looking at content and whether it brings value to visitors is critically important.

This answer depends on the setup of the new site. If it a site is more of a redesign (vs. a rebuild), then typically URLs are maintained. However, when a site is rebuilt, often times it involves new content or the way in which the content is organized (the navigation / site architecture). In this case, it's likely the site will have different URLs. If that's the case, a simple 301 redirect is all that's needed to create the redirect from the old page URL to the new page URL. That way if someone has that page bookmarked (direct traffic) they will land on your new page (or at least new URL). These are critical from an SEO perspective as well, essentially educating Google on your updated URL, and you generally won't see much of a lag if any around positions. However, positions are very fluid so you may see an impact, though not necessarily related.

I highly recommend implementing 301 redirects if you are changing your URL structure. This will help preserve some of the positive SEO signals you have generated. Keep the URLs as is if solely the content is changing. Typically preserving URLs is the recommended approach.

Positioning and messaging are separate from the technical implementation of the URLs. Any time you change your copy, even if for positioning, the content is different therefore your SEO rankings may change.

Focus on delivering helpful and valuable content not content solely for SEO purposes.