Can it cover an entire month? Two?
Completely context-dependent. Try googling "最近の年号" — in this phrase, 最近 goes back a century or more.
Yes it can cover an entire month. And yes, it can even cover two.
People might complain that this answer is offensive, but I feel English-centrism or bias in this question. Have you ever thought of the following questions?: How long of a time period does
recent cover? What is the threshold of length that divides things being referred to as
this or as