Can I issue a request to Google to remove an image search hit that affects our company in a bad way when Googling my company name?
The image in question is not on our website. It's on other website which we do not control.
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Most likely, you cannot. But don't worry — all hope is not lost.
Of course, as suggested in the other answers, you could try simply asking the site hosting the image to take it down, replace it with a more flattering image, or at least to change its name and descriptive text to make it rank less prominently for your company name. However, I'm going to assume that you've tried this already, or at least have good reason to believe that it would not work.
Also, if the image is deliberately designed to unfairly tarnish your company's reputation, or uses your company's copyrighted or trademarked content without permission, you might have a valid legal case, or at least a credible threat of one. On the other hand, you might not — even if the image is clearly intended to harm your company's reputation, and even if it does make blatant use of, say, your company logo, it might still e.g. qualify as protected political speech. We're not lawyers here, and even if we were, it would still be impossible to say anything definitive without seeing the specific image, and the context it's used in, and knowing the relevant jurisdictions. If you think you might have a legal case, talk to a lawyer. That's really all I can say about that.
Assuming neither of those options seems practical for you, your best strategy is probably to bury the unwelcome image as far down the search results page as possible. To do that, you'll need to get other, more relevant and appropriate images to rank better for your company name. You may want to consider getting the help of a reputable SEO company for that (but do make sure that they're really reputable — there are still a lot of bad actors and snake oil vendors in the SEO business), but here's a few quick basic tips:
This should go without saying, but you should start by finding (or creating) some images that you do want to show up in the image search for your company name. No images, no image SEO. Your company logo is an obvious choice here, but pictures of whatever your company makes or sells could also be good.
Make sure images on your company web site have meaningful file names, and consider including the company name in the file name, if it's relevant. Names like
IMG01735.JPG have no SEO value;
widget-a1000-front.jpg would be better;
CompanyName-ProWidget-A-1000-front-view.jpg might be even better yet. See Google's image publishing guidelines for more information.
Make sure your web pages have meaningful
title attributes for their images, as search engines use this information (in addition to file names and surrounding text) to determine what an image is about, and so which keywords to rank it for. (But don't keyword stuff!) In particular, if your web site design features the company logo (and if not, why not?), it should almost certainly have your company name as its alt text.
This should also almost go without saying, but make sure that Google is actually crawling and indexing images on your website, and that you haven't, say, accidentally blocked them via robots.txt. Consider creating an image sitemap for your website.
Look for opportunities to get (appropriate) images related to your company on other high-ranking sites. Does your company have a Wikipedia page? If yes, does the page have good images? If not, consider uploading some yourself, and make sure to choose good file names! (But do first take a good look at Wikipedia's image policies, and especially their policy on non-free content.)
Similarly, perhaps there are blogs or news or review sites that might be interested in writing about your company and/or products; if so, encourage them to include images when they do. You may want to prepare a "press kit" for your company, with clean and up-to-date versions of your logos and other relevant graphical material, and to make sure that your company web site displays a simple and understandable image use policy statement.
Also look for business directories, especially if your business is a local one — many may allow you upload relevant images for your company's entry (but again, beware of bad actors — there are a lot of "directories" out there that are little more than link farms).
Does your company have an official Twitter account, Facebook / Google+ page, etc.? (And again, if not, why not?) Posting relevant images (and, of course, other content) on social media is one good way to get them noticed, both by your customers and by search engines. (Google+, in particular, enjoys an obvious "insider advantage" when it comes to being indexed by Google; that alone is a good reason to have a presence there.)
No, it's not part of Google's policy.
"Right to be forgotten" law (EU) is for individuals only.
For an example, search for "jew", in the front page there is anti-semitic page jewwatch and Google have said they won't remove it either, even though Brin and Page (Google founders) are jews.
If you think someone is spreading lies about your company that are not true or go against the laws in your country, contact your lawyer/authorities. Google won't do anything about it.
If they use your trademarks (logo?) (as part of the shaming image), maybe you can file DMCA request to the server it's being hosted at.
Edit: I'm not sure if DMCA covers trademark infringement, most likely only copyright infringement.