Does AI display actual dimensions of the file?

You can set the size of the Art Board or Canvas.

But vector files are scalable... meaning that you can print them large, super large or small and they do not lose quality.  So basically you could work on an image the size of an A4 piece of paper and scale it up to the size of a billboard and it will not lose quality.

This is why it is preferred to design logos in Illustrator as opposed to Photoshop.

This is the video I share with my students when explaining the difference for a vector graphic and raster graphic.  https://youtu.be/-Fs2t6P5AjY

I'm not clear what you are asking. This merely depends on your document settings, the DPI and how you set the options on the rulers and info palette. If they are set up correctly, AI will display correct measurments right out of the box and honor these settings in input fields and so on as well.

Mylenium