Yes html is all parsed and rendered by the web browser. What the elements do and how they interact and are displayed is defined by a standards body like the w3 consortium www.w3.org/TR/2014/REC-html5-20141028/

There’s traditionally been differences in the implementations of those standards between browser companies, thus causing browser compatibility issues where a site may say it doesn’t work in Firefox, or requires chrome or whatever. Though most major browsers use Chrome’s rendering engine now except for Firefox and its derivatives.

Yes I suppose it is less efficient than precompiling a webpage and serving it as a package that gets downloaded and “executed” though that then opens you up to cross operating system compatibility issues such as Linux and windows not being able to run binaries compiled for the other os. Html was conceived at least in part to be agnostic in that way I believe. As a “hypertext mark up language” it was a way of formatting text for easier reading