Rude to start an email with someone's name?

"Rude" is too strong a word for this. It is informal, as starting with "Hi John" is informal, just in a slightly different way. Neither of them is up to the standard of a very professional business communication. 

From now on, you should always address the person who wrote that complaint with the "Hi John" type of greeting, which they have expressed such a strong preference for. But unless that person has actual authority over you in the chain of command, I think you could continue to address your coworkers (who haven't complained) with just the person's name. 

That's fine to do internally- (within your company), but not when corresponding with clients or other companies outside of yours. Externally, I'd keep things more formal. 

You don't need to start it with anything because the "To" line is already filled in to address the recipient. That's why emails, like memos, don't call for salutations. In fact, many organizations, nearly all large organizations, have established policies barring salutations in the body of emails, barring starting the body of emails by addressing their recipients by name, which policies are generally found in an organization's style guide or communications guide under its provisos for electronic communications. Any business writing class will tell you it's a faux pas to start emails such a way.

Moreover, if you simply start by addressing the recipient by name, it feels patronizing, like you're setting out to manipulatively talk down to them, or, worse, like you're about to scold them. That's because in normal conversation, starting by invoking someone's name when you've already got their full attention, like you do when someone's sat their reading the body of your email, is what someone does when they're about to patronize you or what your mom or a principal does when you're in trouble and they're about to lay into you. 

Remember, emails don't have the benefit of conveying your actual tone of voice or your body language, so starting out such a way can readily cause your email to be read in a tone very different than what you may have intended, and since it is you who are writing it, it is you who must be careful and take measures to ensure your tone is clear. So unless the tone you're going for is one of being patronizing, which is rude, or scolding, which also can be rude if unwarranted, then it is incumbent on you not to start your emails in such a way. 

It's not at all "rude" to do this in-company. But it would be unprofessional to do it when sending emails to clients.

I would tend to think no. We live in an age wear blue jeans and shorts are replacing suits. I would refer to the policy of your company if there is one and just stick to that for email standards. 

I'd ask people at work. People on Yahoo answers aren't in your workplace let alone your field in all likelihood. I think it makes sense to get additional opinions on this, but just not from us. People are answering based on considering a hypothetical, and there's additional factors that are local to your work that we can't speak on. 

No, why is it rude to address someone's name to start with?

From what I could find out from my limited research, a professional email should start with either "dear," "hello," or greetings" is acceptable.

If you know your coworker by the their first name, that is perfectly appropriate to address them the same way in an email.

It’s not great etiquette but it can be acceptable, it shouldn’t be taken as an automatic slight 

i dont think so