I have two questions regarding the Joomla content editor image tab and the file browser.

First: Is there any way to add multiple image sizes using the srcset functionality in the file manager? This functionality works on the Image Manager Extended responsive tab, but not on the file manager as far as I know.

The Images & Links tab is part of the Joomla Article Manager, and those image fields are extended slightly to provide access to the JCE File Browser to select images. Setting the srcset attribute for responsive images requires extra input, to specify the width descriptor or device pixel ratio, which is not available in the JCE File Browser, and cannot be added as additional options in the media field itself.

Second: Is there a way to get the image's width and height attributes from the file browser and display them in the HTML for intro and full text images?

Please note that the rendering othe Intro Image and Full Image fields is not done by JCE but by the Joomla Article Manager via the Joomla Media Field. This field does not accept additional attributes for the image tag, so the width, height and srcset attributes could not be set, even if the data was supplied.