Grammar study is necessary because languages shape the way we think. If your language lacks a subjunctive mood, then your ability to express doubt in the way native speakers of your L2 do, going in, will be retarded.
Same for users whose native language lack progressive tenses, etc.
This goes both ways. Anglophones learning other language encounter similar issues (Cases/Inflections, Gender, Word Order, etc.).
This is why we can often tell what a person's native language is based on their grammar. People who think they can become native-level simply by ignoring grammar and going the immersion route are in for a rude awakening. We see the results of this in the US (and perhaps all other countries) all the time. People are here for 10 years, and speak fluently, but with deeply ingrained structural faults in their language production. It doesn't work that way.
How we learned our native language doesn't matter. The fact that we have a native language is what matters. Babies are a blank slate. The native language they learn, and the linguistic, social, and cultural environment in which they are bought up shapes the way they think about the world. No one can reproduce that (short of a TBI that causes them to "lose" their language), so it's useless to even bring this up...
Yes, this can be achieved through immersion. But, a dedicated student can probably master this through study just as quickly (and probably quicker) than the average 30 year old trying to assimilate it primarily via immersion.
Learning more in 3 months through immersion than 10 years of instruction isn't really an indicator of how great immersion is. It's an indicator of just how bad the instruction is. It can also be a side effect of the lack of need to use German as a primary language (so the language was largely learnt, just not really activated until you felt you HAD to use it on a daily basis).
Classical Music is very good for language study.
Movies & TV are bad for beginners. They won't understand anything, and it's a wasteful way for training comprehension, IMO (the amount of hours it takes is unreasonable compared to graded materials) - unless the material is designed for learners.
Find a decent method (Colloquial and Assimil are a good combination). Get some Graded Readers in eBook and Audiobook format. Read while you listen. If the audio is too fast, slow it down to 0.75X :-)
Practice Phonetics and Intonation first. Dedicate a couple of weeks to it, IMO. How you pronounce and speak the language dictates what your brain expects to hear when you are listening to others speak. Having good diction is going to eliminate a lot of potential issues re: comprehension, speaking, and even writing/spelling (to a lesser degree).
I've never found Flash Cards to help me, much. Most methods or grammar books integrate vocabulary acquisition. I think if you read a lot of graded material, that your base vocabulary will come quickly and you will acquire vocabulary quite easily.
Flash Cards tend to feed you a lot of vocabulary that you simply don't need at the moment. You can only learn so much at a time. Keep it personal and basic at the beginning.
Once your base vocabulary is strong, it will actually enable you to learn more advanced vocabulary more easily. This is especially true for German, due to its tendency towards compound words.