What is the next number in this matchsticks series riddle?

Can you solve below mathematical equation?

2^1234 - 2^1233

Can you write down eight eights so that they add up to one thousand?

It's a 7-letter word.
If we remove 1 letter from it, it remains the same.
If we remove 2 letters from it, it remains the same.
If we remove 3 letters from it, it remains the same.
If we remove all the letters from it, still it remains the same.
What is it?

What brings you down but never up?

A King wants to send the diamond ring to his girlfriend securely. He got multiple locks and their corresponding keys. His girlfriend does not have any keys to these locks and if he sends the key without a lock, the key can be copied in the way. How can King send the ring to his girlfriend securely?

The average age of 10 members of a committee is the same as it was 4 years ago, because an old member has been replaced by a young member. Find how much younger is the new member ?

If all the below statements are true

1. Gianni was either in Italy or France in 1997.
2. If Gianni did not kill Versace, Hilton must have killed him.
3. If Versace died of suffocation, then either Gianni killed him or Versace committed suicide.
4. If Gianni was in Italy in 1997, then Gianni did not kill Versace.
5. Versace died of suffocation, but he did not kill himself.

Who killed Versace and where was Gianni in 1997?

Complete the given grid with valid words. You can only use the letters AAEEIIMMPPTT.

Hint: The grid reads the same across as down.

People put me on the table and cut me but do not eat me. What am I?