国語 means the Japanese language in that case.
辞典 and 辞書 are basically the same. According to 日本国語大辞典, 辞典 is a new name of 辞書 (the first use of 辞書 was in the late
Edo period, and the first use of 辞典 was in the early
Meiji period). They say that 辞典 is preferred as a suffix (e.g., 国語辞典, 英和辞典), and 辞書 tends to be used as a single word (e.g. 辞書を引く).
天の河 and 天の川 are the completely the same; "Galaxy" as a natural object visible in the sky.
銀河 can mean 天の河/天の川, but it's more common to be used as a general term "galaxy", I think.
銀河系 refers to "our garaxy"; the galaxy where our Solar System belongs to. 天の川銀河 has the same meaning.
Thus, when you look up the night sky, you can say 天の河/天の川/銀河がきれいだ, but can't say 銀河系がきれいだ. Or, you can say アンドロメダ星雲は隣の銀河だ, but can't say アンドロメダ星雲は隣の天の川だ.
火垂る is a hypothesis about the etymology of ほたる.