In my opinion a logo can be any size as long as it fits your website and everything. File size well there's ways to lower image file size optimized for web pages.
Completely dependent on its usage. There is certainly no general size a logo should be. If you're asking about the image file size then I would simply test it. The lower the file size the faster the loading but make sure you do not lower it to much as to affect the quality of the logo.