Please feel free to correct me and teach me if there are better expressions

We would not use "weak" in this case.  You could say:

I don't feel well

I feel queasy (nauseated)

2. I’m a bit unwell again.. I have a dull stomach and a headache.

You can use "dull" for headache, but not for stomach issues.

I have a stomach ache and a dull headache.

3. I guess I have a cold coming

Normally, you would say: I have a cold coming ON.

4. I've always wanted to know it exactly!!

I've always wanted to know exactly that (specifically, "that thing")

5. It was strange I dreamed a lot last night. But I wasn’t able to remember what happened like everybody else.

It was strange: I dreamed/dreamt a lot last night, but I wasn't able to remember what happened in my dream/wasn't able to remember the dream like others/other people do