Instead of copying and pasting an jpg/png image, is there a way to create a post (within a thread) of a live website exactly identical (with live, functional buttons where when clicked will open a separate browser)?
XenForo's BB codes allow for recreating functionality like inline images and hyperlinks. Hyperlinks to external pages always open in a new window. But in general you cannot recreate web layout (styling and positioning) using BB code. Using XenForo's WYSIWYG editor you can copy from a web page and paste directly into a post. The editor will automatically convert what it can (mainly images, hyperlinks, and text styling).