Everyone says there is no difference between "site" and "sub-site". This makes all these "how-to" articles useless since they don't answer the basic question: can I create new page layouts, master pages Just for the Sub-site without breaking anything on the already existing "root" site? If the answer is "yes", can you please explain how-to do it? I need to create a sub-site of the existing "root" site, the sub-site with its own layout.
For page layout, we cannot create the page layout just for sub site. When we create the page layout, the page layout will be saved in the page layout library in the site collection.
We can configure which page layout can be used in the subsite. Go to site settings->Look and Feel->Page layouts and site templates.
For master page, we can create the master page for the sub site in SharePoint designer.
And you could go to the master page gallery of the subsite by the URL below:
Of course you can. BUT You'll need the Publishing Feature activated on all Sites and Sub-Sites. Otherwise you're not able to access the needed configurations. Once you activated the feature and published your (masterpage and) page layout, go on Site Settings in the root web and there to "Pagelayouts and Templates" (or simly add "/_layouts/15/AreaTemplateSettings.aspx" to your URL). Check if any disturbing configs are there which may interrupt the inherit of templates and layouts to child webs.
If all's fine there, go to the wanted sub-site, and do the same here. You may have to do some config work which allows you to use all or just a specific subset of layouts. That's all. But keep in mind, that every (and I mean really EVERY) Sub-Site that shall use your custom layout, must have the Publishing feature activated.