please check this conversation.

Jack: You haven't done it yet. What have you been doing for the last week?
Alex: I've been sick the last three days. Otherwise, I would have done it.

tenjingHere the use if "I have been sick for three days" is correct and also the conversation? ?

Yes, but I've changed it a bit.

I have two questions here. One is that I must use 'past' with period of time or it is optional? Another question is that Alex is no longer sick now right?

tenjingOne is that I must use 'past' with period of time or it is optional?

You can use either the present perfect or the simple past.

1. Alex: I've been sick the last three days. Otherwise, I would have done it.
2. Alex: I was sick the last three days. Otherwise, I would have done it.

ten

So,we don't need to use 'doe's in sentence 1? http://www.englishclub.com/grammar/verb-tenses_present-perfect_u.htm
It says, "he has been ill for two days" means he is still ill.

So confusing, please check this. http://www.englishclub.com/grammar/verb-tenses_present-perfect_u.htm

tenjingIt says, "he has been ill for two days" means he is still ill.

It is most likely that he is still ill. It will depend on the context.

We use the present perfect in conversations like this:

I've been sick the last two days, but I am much better now.

What about this sentence?
A. I have known her for the last ten years. It means I still know her right? ?

tenjing What about this sentence?A. I have known her for the last ten years. It means I still know her right? ?

Yes.

tenjing I have known her for the last ten years. It means I still know her right?

You can become well after an illness but you cannot 'unknow' someone.

Mister MicawberYou can become well after an illness but you cannot 'unknow' someone.

Sadly, my beloved aunt had a severe case of Alzheimer's disease and she "unknew" everyone except her one and only daughter. So it is possible to "unknow", but then the "unknower" would have no memories or claims of having known.