'Over' here means 'about': Xi's message was about Tran's death. Similarly you could have a dispute or a fight over something (e.g. the US and China currently have a dispute over trade). It is a slightly unusual substitute for 'about', and it is quite hard to think of cases where it applies apart from these. Don't confuse this with the phrasal verb 'get over' which means 'recover from'. You can get over something like an illness, a disappointment or a death; the news article you quoted did not describe recovery.

It is a fixed expression, so yes the grammar is correct. From the way I see it in this context it means "in regards to the death of.." You see this very commonly when someone dies, but more so in this context- "John can´t get over the death of his father" Which means he is not able to feel better, that the death of his father is still affecting him. Hope that helps, -Mark