Generally, no matter where you are, the answer to your question(s) is largely going to be a fact dependent analysis. Just so we're clear, I want to be sure you're using the term pseudonym in the way that it's generally accepted or understood to mean, being a fictitious name typically used by an author (this could be book, blog, a paper – whatever); but not as an alias or a false identity used to mislead people about your qualifications or official position.
Assuming you are using a pseudonym in the way it would typically be used: as a mechanism to write and publish under a name other than your own so as not to have credit, or discredit be given to your actual identity, than I cannot see this being a problem anywhere, nor could I find any laws that pertain to this. (You've not asked about copyright/trademark, which poses different legal issues but is also not insurmountable if done correctly.)
Generally, if you are publishing something, you have the right to publish anonymously or use a nom de plume (aka:a pseudonym or pen name)...this has been done throughout history and throughout the world by many famous authors. Reasons differ and the examples for such are endless. I have heard of a famous "pulp" fiction writer having used one when writing a children's books; people do it to overcome biases and exposure associated with their name (both positive and negative); during WWII Jewish authors used them to avoid being discounted or arrested based on their name and lineage. In the 1700's (when the U.S. was coming into its infancy) many writers and journalists used pseudonyms, and while the use itself wasn't illegal, they did so to write controversial and often illegal articles, papers, and even letters to the editor, that criticized governments or monarchies and their practices. Some of the most famous pseudonyms were for that purpose. Ben Franklin used to almost exclusively write under a pen name and create elaborate characters to go along with them. Famous authors like Dean Koontz and Stephen King, both used pen names when trying out new genre's or to avoid over-exposure – especially when releasing at the same time. Many famous women authors, throughout history, used to write under male pen names because women couldn't be published or at least not as easily. Most famously was probably the Federalist (Federalist Papers), written by James Madison, Alexander Hamilton, and John Jay (under the pseudonym Publius), who published 80 or so papers calling for the ratification of the United States Constitution. These were very dangerous writings for their time.
When I read your question, however, I worry that you seem to be suggesting something other than simply a pseudonym for these various typical purposes. Maybe I'm wrong, but when you say hide behind, and you are talking about using Ph.D., or Dr., or some other title that would denote special education or training, you are treading into different territory if the purpose is to mislead or misinform people: not about who you are, but about what you know based on who you are. There is a big difference.
So, for instance, Dr. Seuss (who actually began his career writing and drawing political parodies and cartoons) wrote children's books using Dr. as part of his nom du plume, which was his prerogative as nobody was mislead or harmed by the fact that he chose to put Dr. in front of his name. He was not posing as a doctor or giving advice about medical issues, so nobody detrimentally relied on his writings based on that fact. However, if someone wrote a non-fiction book about life saving holistic cures for cancer (I'm just pulling topics out of my head), and the author called himself Dr. Doe, or you write a book about prevention of suicide and you put Ph.D. after your name, and people who read the books detrimentally rely on the information because you've mislead them about your expertise – that is not just a pseudonym and it could definitely leave you vulnerable to legal problems. You cannot hide behind a title to exploit people by suggesting you're qualified to proffer information in which you have no training or expertise; this is not a nom du plume, it is a fraudulent misrepresentation.
With regard to whether or not you have a right to hide behind a pseudonym on the internet? E.g., assuming I don't break any laws, can an internet host/provider publish my real name? that really depends on the site you post on. It is your responsibility to read their privacy disclosures to find out. Facebook, for example, has a "real name" policy, which disallows this practice. Other sites may allow you to create any screen name you want, but require a real name for registration purposes for a variety of reasons, one being if they need to have it for law enforcement purposes. Regardless of the policy of the web host, if you do break the law, or end up getting sued based on your writings, all it would take to reveal your identity is a subpoena arising out of a lawsuit.
While using a pen name is seemingly legal in and of itself, pretty much anywhere, whether or not doing so could lead to criminal or civil liability, or whether your real identity could be exposed, is largely dependent on how you undertake to use your fake identity. Impersonating certain licensed professionals, or government officials can be a very slippery slope, and could be illegal or lead to liability, in an of itself. Personally, I would stay away from using any.