Let's address each question one by one, starting from the top.
I'll start with your second question: What is an extremist? We can start with a dictionary definition. From Merriam-Webster: "1. the quality or state of being extreme. 2. advocacy of extreme measures or views." But what does the word "extreme" mean in a political context? We can consider the following definition from Cambridge: "someone who has beliefs that most people think are unreasonable and unacceptable." However, this definition is majoritarian, meaning that a majority of a population can dictate what is considered extreme. In other words, the standard is relative, not objective. Herein lies the problem with defining extremism: it's a vague term subject to the whims of the ever-shifting Overton window of politics.
The closest definition I can offer is the following from Reinhart Koselleck and Giovanni Sartori. I will paraphrase since it's very long. Basically, the scholars argue that extremism is characterized by an anti-establishment mindset grounded in a "struggle for supremacy" as opposed to "peaceful competition between parties...advancing the common good." Extremists stand "at the periphery of societies" and seek to dominate politics by creating an "us vs. them" mentality against other social groups. Extremists tend to be intolerant, militant, violent, and fanatical, where they seek a homogenized society with authoritarian traits and show a disdain for "majority-based democracy, the rule of law, and human rights" (source: https://www.jstor.org/stable/pdf/26297896.pdf?refreqid=excelsior%3Adef7e80258c4aabc3d7b4a4ff795e11a&ab_segments=&origin=).
Given this definition, I think it is fair to say that Breivik was an extremist in every sense of the word. In his manifesto, he praised violent extremist and militant groups such as al-Qaeda and the Scorpions in Serbia. Additionally, he had an affinity for authoritarian leaders, admiring leaders such as Russian President Vladimir Putin (Prime Minister at the time of writing). He admired state-sanctioned violence against Indian Muslims and called for the mass deportation of European Muslims. Breivik has also cited support for Neo-Nazi, anti-Semitic, and white nationalist views.
Moving to your third question, Breivik is usually categorized as far-right extremist because his views align with far-right politics, which is characterized by fascism, ultranationalism, authoritarianism, nativism, apocalypticism, and exclusivism (e.g., racism, homophobia, Islamophobia, chauvinism, sexism).
And finally, are there any far-left extremists? A definition for far-left extremism is hard to come by, but we can safely say that far-left extremism is generally defined by virulent opposition to contemporary capitalism, support for a global revolution or revolutionary action, anarchism, and affinity for violence against elites, government institutions, and far-right extremists. Of course, there are problems with this definition, since many far-left groups disagree wildly on everything from support for violence to support for a global revolution. You would be correct in identifying communists as far-left extremists. Concrete examples of far-left extremism (and terrorism) include the Red Army Brigade, Shining Path, Japanese Red Army, FARC, and the Red Army Faction. Similar to the far-right, these far-left groups create an "us vs. them" paradigm (albeit with different groups), advocate for violence, and shun the democratic process as an effective means for advancing political goals.
