News has lifted my spirits - THE (or “this”) news has lifted my spirits.
Did you save me a slice of that cake? - this is fine
The moment of reunion was very touching - “the reunion was very touching” sounds more natural. Technically your sentence wasn’t incorrect, it just doesn’t sound natural.
I’ve gotten used to living here - fine
You don’t care whether I’m okay or not - also fine
Anyone who’s back home knows that - “anyone back home knows that” also works and in my opinion, flows better. But either way, it’s just fine.