To start with - No. English is actually an easy language to begin learning because:
This is a better question, but it's huge. So, we've written a whole section on how to improve your English here.
There's no "have to", but you probably won't be able to progress beyond reasonable proficiency without it. That said, when you begin to learn English, speaking and building up your vocabulary is more important than worrying about correct grammar. However, as you progress you will need to incorporate more grammar into your learning programme
There is a common impression that learning English grammar is painful, but it is a lot easier than many other languages. Of course, English is more than just memorizing grammar rules, the grammar should be incorporated into your everyday use of the language, theory and practice should go hand in hand.
Think of English vocabulary as the bricks of the language and grammar as the mortar that hold those bricks together, without the mortar the bricks can come tumbling down, and a lack of a basic understanding of grammar can be embarrasing. At an advanced level you should be thinking much more about the correct use of the English language. And eventually you should be able to forget about it all together and start doing what comes naturally.
Generally English grammar is not as difficult as some other languages, such as German. We have the French to thank for that. In 1066 William the Conqueror brought to England the French language as well as the ability to build impressive castles. French was the official language in England for nearly 300 years and so the English language was kept alive by the common people (rather than the academics who do like to keep the status quo). As a result nearly all the difficult word-endings, inflexions and marks of gender and case were gaily dispensed with.
You don't expect me to answer that, do you? Actually - it depends. It depends on which English you will be most exposed to. If you are moving to the US then learn American English, if you're going to work for a British company then learn British English. And don't forget there are many different kinds of English and the rising star of International English.