I think you have the right idea- nouns and na-adjectives require a copula (だ・です) before it. That way, it doesn't get mixed up between the "from" meaning of から.
しずか ですから 散歩しましょう
Correct; a copula is required if a na-adjective comes before から.
But I do want to point out that you can still use だから (as opposed to ですから) even if the sentence is polite. The most important thing is that the main (i.e. final) verb is in the polite form.
Correct; with い adjectives using だから is ungrammatical. (You actually can use ですから with an い adjective, but it isn't necessary.)
You are correct that you'll need だから or ですから after the noun こと. But I don't see any reason to have こと there at all. It should just be 飲むから (or, in this case, 飲んでいるから).
Also, the inconsistency in politeness here is a little strange. You used impolite -てくれ at the end, but polite ですから in the middle. It is weird to have an impolite independent clause with a polite dependent clause. But that's not what this question is about.
Correct; だ or です do not (and cannot) immediately precede から if there is a verb and not a noun or na-adjective there. (However, this sentence also has the same problem as last time with the inconsistent politeness)