#1, #2 and #3 all look OK to me. The subject of #1 would most likely have to be a president or similar, or the personification of a country.
I think you “mourn for” something you miss, so you cannot “mourn for losses”.
The simple answer is that “for” is omitted in #3 because “mourn” is being used transitively.
The usual idiom is “rain cats and dogs” (no “like”).
#6 sounds a bit odd to me. “forefathers”, plural, would be natural, and the usual idiom is “follow in someone’s footsteps”. Anyway, I understand the sentence to mean something like “Do they conduct themselves in the same way as their predecessors/ancestors?”