Let's say I own an LLC, and I'm also starting a website. In case of legal troubles with the website, I want to be free from personal liability. Is there anything special I have to do to link the website to my LLC, so it's a "product" of the LLC?
An LLC isn't a magical way of avoiding legal liability. If someone enters into a contract solely with your LLC, then you might be able to avoid liability under the contract – which is why banks invariably require loans to LLCs to be personally guaranteed. But if you commit a tort or crime as part of your LLC's business, you're still on the hook.
Max Kennerly, a US trial lawyer, explains this in a blog post: Can I Set Up An LLC To Avoid Personal Liability In A Lawsuit? He cites the following passage from Restatement of the Law, Third, Agency § 7.01:
An agent is subject to liability to a third party harmed by the agent’s tortious conduct. Unless an applicable statute provides otherwise, an actor remains subject to liability although the actor acts as an agent or an employee, with actual or apparent authority, or within the scope of employment.