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There is no a page layout for Page Content type, it is by default. You can refer to the following screenshot about the default Page Layout content types:
When you click Page via New Document->Page, it will redirect to CreatePage.aspx, and you must to choose one Page Layout to create.
Assuming you choose "(Article Page) Body Only", after you create it, you ...
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You have to use WebId property in webpart snippet to get data from specific subsite. By specifying webid property it will try to find Pages library from web id and display data from subsite rather than Site collection. below is updated snippet on how to display from subsite.
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