Thanks for the examples. You are right, は can obviously be used to introduce something new as well.
As for the 3 different が:
1. Conjunction: yeah it is a bit confusing, but we have to live with the fact that the subject marker and the conjunction coincide in form. There are also words like these in other languages as well. For example, the word "general". It can mean "general" as a rank in the army, or "general" in the meaning of 一般に or maybe 大抵.
2 and 3: I always think of the が used after an object like of a subject
e.g.
私はそばが好きだ。
I try to think of this sentence not as "I like soba" but "As for me, soba is desirable". In this case soba is the subject of the sentence.
