Writing is not necessary for fluency. Some language groups have no written language at all. Deaf people are usually fluent in sign language, yet there is no written language of sign. Children who are too young to read and write may be fluent for their age expectations. And as Joe S. points out, many people throughout time have not been literate but have still been fluent in one or more spoken languages. Likewise, one may be a fluent reader of a language without being able to speak it at all. Again, I would use Deaf people as an example. Many can read and write English fluently; and yet just as many will never speak it aloud. This is true of other languages as well.

Unquestionably. According to the World Bank, the current global literacy rate for all people aged 15 and above is 86.3%. If you go back a few hundred years, perhaps some 10% of people could read or write. Yet illiterate people are not only fluent in their own language, but commonly speak others depending on their region. So, in short, yes. It is possible to speak another language without learning how to write.

Yes, but you will be limiting yourself.

Of course. I would submit that most native speakers of all languages can’t really write properly in their respective language.

Yes, definitely! I have met many people who speak well and write little if at all. It can definitely be useful to learn how to write, but it’s better to only be able to speak than to be able to do nothing. Honestly, I could see myself only learning to speak if I were to learn a character language like Chinese. The writing system couldn’t be more different from the languages I know, and I’m not good at memorization.