Nope! Learn the most common one. Some synonyms can be difficult or rare, so focus on ones that you actually saw before. If you find two words that mean the same, ask people what the difference is. However, when you're more fluent, use synonyms to get a better understanding of what words mean (by using thesaurus) instead of using a dictionary.

It is important to know them as passive vocabulary. You need to be able to understand them when you read or hear them. It is not important to use them yourself. Advanced writers will try to find "just the right word" to express their meaning. However, George Orwell has written "Never use a long word where a short one will do" and "Never use a foreign phrase, a scientific word, or a jargon word if you can think of an everyday English equivalent."

I agree that in the beginning you should learn the most common words. But as you make more progress, synonyms are very important. Firstly, they are important because they help you to add variety to your vocabulary in order to sound more fluent, and to write better. Secondly, English speakers don't like to repeat the same words over and over again when they are talking about something, so they use synonyms. If you learn synonyms, you will understand native speakers better. Finally, some people prefer to use certain words, while others prefer synonyms to these words. For example, some people say "He's acting crazy," but I might say "He's acting silly." In this case, "crazy" and "silly" are synonyms.

I believe it's important to learn synonyms of words, as for example if someone uses petite for small you would not understand what this means. However, it's is fine enough to just learn the most commonly used word as you learn English.

You mean the meanings of words? The ability to use another somewhat similar, if less common word, for variety or accuracy, or to convey a slightly different shade of meaning, texture, or emotive reaction, when you want to, rather than be limited/constrained to just a few scant, limited words that you already know? . Important for what? Depends on your purpose, I guess, speculate, summise, ponder. :)