It is important in the sense that you want people to understand what you are saying best. But you can have bad grammar and still be roughly understood by others also. It depends on how serious you want to learn the language. On the other hand, even native speakers are lax on their grammar sometimes during casual conversation.

Very important in most European languages including English. An improper usage of words often may make a sentence completely incomprehensible or pervert its meaning.

In English, I think the importance of grammar depends on where you live. Lots of Americans make grammatical mistakes during a normal conversation, but since we are native speakers, we can still understand what the other person is trying to say. (For example, a LOT of people say things like, "Me and Bob went shopping today." even though it is grammatically correct to say "Bob and I went shopping today." "I" is supposed to come last when listing things that include the speaker, but lots of people forget this. This is only one example, but the misuse of grammar in the US is widespread.) For foreign speakers, I believe proper grammar is very important. Using proper grammar is seen as a polite gesture, for native speakers AND foreign speakers. (If you were speaking with your boss at work, or a really respected individual, you would definitely need to use proper grammar.) I can't say this is a fact, but if you use proper grammar, you could probably avoid some racist comments. Some people in America can be quite rude when it comes to foreigners, and most of the time it is because of the broken English. If you speak with proper grammar, you will be perceived as an intelligent person. Anyway. Sorry about this long-winded response. I hope it was helpful!

yes, grammar is very important