If we define "studying grammar" as the formal act of sitting down with some heavy books and trying to memorize rules then yes, by all means, I do believe that we can learn a foreign language without studying grammar. But I also call noticing certain patterns after a lot of listening or reading in your target language and applying them in your output "studying grammar" therefore I have to go with the "no" option.

Grammar is a good tool to learn, though I think you can learn any language by listening but not just studying grammar. I think grammar is useful to understand certain things, but not to learn them the way you need to speak a language fluently. Maybe most advanced Spanish learners know Spanish grammar even better than I do, but grammar only says how a language is spoken by native speakers, so you can also learn grammar from native speakers who speak properly.

I think it is possible but it will be a very slow process. I dislike the "learn like a child" method haha, but that`s what your question implies. Children don`t bother about grammar, they only listen and imitate etc and they learn it, even better than a billion adults would but children have a lot of time to develope their language skills and actually we never stop learning our mother tounge. So, yeah I bet you would learn it, maybe even better than any of us, but it will take a loooooot of time o_O

Sure you can. And you can speak it as well as a native speaker--if you learn it as a child, usually before the admittedly flexible "critical period" of roughly 12. But learn it as an adult without grammar study=a really awful sounding pidgin language. I've met a lot of cats that haven't studied much grammar and have lived a few years immersed in the language that think they speak really well and it's quite awful.

Till certain level , I think . But to keep going in learning it , you'd need to study grammar :)