InDesign doesn't make a cover html file for you. You have to make that yourself if you want one. But if you are just doing mobi, you shouldn't include one anyway and KindleGen removes it (I think that's new, isn't it?).
Designating the cover image in InDesign is done for other reasons like telling the iTunes store what to use. It puts a meta name="cover" tag in the metadata section of your .opf file, adds the image file to the manifest and packages the image up with the epub.
Speaking of KindleGen, it is built into Kindle Previewer so you can just drag and drop you epub file onto the previewer and it will convert it to mobi. No need to use Calibre.