Hi,
No, your website will work just fine without a robots.txt file.
Without a robots.txt file search engines will have a free run to crawl and index anything they find on the website. This is fine for most websites but it’s really good practice to at least point out where your XML sitemap is so search engines can find new content without having to slowly crawl through all the pages on your website and bumping into them days later.
It shouldn't go to homepage if mywebsite.com/robot.txt doesn't exist shoud go to custom 404 error page.
Hope this helps.
Thanks