How much does knowledge of linguistics help with learning foreign languages?

[Javaman]

So, I have two related questions for you. I think no previous knowledge of linguistics is necessary for the discussion.

1. Does the knowledge of regular sound changes help with learning a foreign language related to a language you already know?
For instance, let's say you know English very well. Does it help you with learning German to know the following and similar rules (they apply to the Germanic words and very early Latin borrowings):
English "t" corresponds to German "z", which is read "ts", in the beginning of a word (two-zwei, ten-zehn, tooth-Zahn, tongue-Zunge, twig-Zweig…) and after "r" (heart-Herz…), but corresponds to "s" otherwise (water-Wasser, it-es, that-das, what-was…) except after "s", when it corresponds to "t" (star-Stern, stone-Stein…)
English "th" corresponds to German "d" (three-drei, that-das, thick-dick, mouth-Mund, death-Tod…)
English "v" corresponds to German "b" (seven-sieben, give-geben, live-leben, have-haben…)
English "d" corresponds to German "t" (desk-Tisch, word-Wort, god-Gott, ride-reiten…), but English "nd" corresponds to German "nd" (wind-Wind, hundred-hundert…)
English "oo" corresponds to German "u" (book-Buch, foot-Fuss, too-zu…)
English "one" and "ome" corresponds to German "ein" (pronounced ahyn) and "eim" (pronounced "ahym"), respectively (one-ein, stone-Stein, alone-allein, home-Heim…)
English "ou" corresponds to German "au" (house-Haus, out-aus, show-schauen…)
English "ea" usually corresponds to German "o" (ear-Ohr, east-Osten, easter-Obster, bread-Brot, bean-Bohne, death-Tod…)
I believe this was enough for anyone to get the basic idea.

2. Does knowing an archaic language from some family help with learning modern languages from that family?
For example, I've heard that, if you know Old Church Slavonic, you can basically understand all Slavic languages. Is that true?

I'd say yes and yes to your questions.

My follow up question would be... What matters more: phonemic similarities or the underlying syntax and grammar?

My (very limited) understanding of linguistics is that languages are grouped moreso on syntax and grammar than phonetic similarities. Phonetic similarities are more a symptom than a cause. I'm happy to be corrected if I'm wrong on that.

On a side note, I find these kinds of discussions highly interesting. I want to learn more lol.