“last” and "past" have different meanings.

On the one hand, if you use "the last/past week":

1. The original version of "past" is "pass", so "in the past week" means "the week which has just past" compared with the time when the speaker is saying these words.

2. While “last” means "something happens at the end of a period of time", so you can use "the last week of April/may/my holiday" to refer to "the week at the end of that month/holiday, with no other weeks left". Based on this meaning, you can use "the last week" to refer to "the week nearest to the present time"

On the other hand, you can use "in the past" to refer to "the time before now",but you cannot use "in the last". There is an usage of "to the last" means "until the end of an event or somebody' s life", where the emphasis is on the meaning of "till the end".

I don't think there is much difference between "past week" and "last week".

And you are right that you can say "I learned that in the past" or "I t was good at Latin in the past".

There are other differences but it's hard to think of them offhand. If you give more examples I can tell you if they are correct or not.

what 'bout (in the last)  at the end of a sentence ? like "this place used to be beautiful in the last"

I don't think this is correct