Were you ever woken up by a noise and couldn't sleep any longer?
Yes. That has happened several times, annoyingly. Boyracers, fireworks and all manner of other things have disrupted my sleep before.
Did you have ever stay a whole night up?
I have had sleepless nights on several occasions. The most recent occasion involved me finishing a bunch of uni work, which I knew was a bad idea since I had a six hour lab session starting the following morning (though somehow I still managed regardless, but I did need to just crash for several hours once I got back to my accommodation).
Can you sleep well when you are at another place far from home (hotel for instance)?
I guess it really depends. I don’t get homesick, but if the environment is especially unpleasant for whatever reason, I won’t sleep well there.
Can you remember your dreams when you wake up?
I can, but usually they are uneventful and I forget them maybe like a day or two later.
Edit: oops, missed one:
Have you ever had such a fatigue being able to laydown on bed and to sleep in no time?
Yes - many times. See my answer to the question about whether I’ve stayed up whole nights before.