– Should I or should I not change site url in settings from http to https?
– Should I install flexible ssl loop plugin?
– Should I install the official plugin from cloudflare? (The server does not have cloudflare mod installed)

On one wordpress site I changed url to https and installed flexible ssl loop plugin. Working fine I guess.

To completely rewrite your site’s url to https, use any method specified in this article. https://kinsta.com/knowledgebase/wordpress-search-and-replace

You don’t need to install any other plugins for this purpose.
Optionally you can setup a page rule on CloudFlare to support automatic redirection to https.

Hope this info serves you well.
Good day!