I have been coughing for the last week, to me, means that you have been coughing for a week and you still are.
If you want to indicate that you've stopped, you might say: I was coughing for the last week, but I'm better now.
It's useful to remember that language is for communication and there is no reason to put too much pressure on a single short sentence to get your message across -- especially when it leaves you wondering yourself. Just add a few more words -- "but I'm better now" -- and put everyone's mind at ease.
Although I have coughed for the last week also has that sense of coughing for a week, I'm more inclined to use your first sentence. The continuous tense of have been coughing is very useful here to get that meaning across.
One last thing: you don't need "past one week." We understand it's one week without the "one."
Monkeyaug, that sentence sounds strange to me. Have coughed seems to refer to a single instance of coughing, rather than a bout of coughing. But I wouldn't understand the need to use the perfective here: I would expect I coughed badly if that is the meaning. If the meaning is that there have been many instances of coughing, you need to use the progressive (continuous) have been coughing.