The difference is in the future of the sentence. Last implies nothing else will follow. It's the last, and after this it is finished. Latest implies that it is the last to date, which means there could be more to follow. The examples in J.R.'s post fit the case: they leave their last will and testament. They won't be able to leave another will after that. These are their last wishes. the ...
In this context, last is a relative term -- it describes the most recent event that would fit. For example, one might say I saw him last Tuesday. Here, last indicates that the Tuesday you're referring to is the most recent previous one, not some other Tuesday. If you are using a full date (month, day, and year) then last is not necessary. Your service expired last June would make sense; your ...
Without a preposition "the last|past three years") would probably would be heard in casual speech more often than it would be read. Steady improvement in cash flow over the three year time-period seems to be the main idea in the statement, so it would be best to make that idea as clear as possible by choosing the best preposition, the best adjective, and by subordinating everything else. "Cash ...
Difference between (in / for / ∅ / during / over) the (past / last ...
The last time you spoke to him is obviously in the past, but it is in the past now. So when is the last time is correct. However, the last time you spoke to him was a time before now. Therefore, When was the last time is also correct. When was the last time is presently used much more often, as this nGram view shows.