Which English verb becomes past tense just by rearranging the letters?

Solve the riddle below:

If a Chicken says, 'All chickens are liars" is the chicken telling the truth?

All of you like to eat me. I am a five-lettered word. Remove the first two and I turn into an infamous animal. Remove just the first and I become a heinous crime. Remove the first and the last and you can groove on me.

Can you guess who am I?

What is the fewest number of coins that would be required in order to make sure each and every coin touches exactly three other coins?

I have branches, but no fruit, trunk or leaves. What am I?

What does this rebus picture identify?

I am a type of vehicle that has two wheels and is powered by humans. What am I?
Hint 1: You can ride me for exercise.
Hint 2: I am environment-friendly.

A pack of cards has 40 cards. You are blindfolded. Out of 40, 25 cards are facing down while 15 are facing up. You have been asked to divide this pack of cards into two decks - so that each deck contains an equal number of face-up cards. Remember, you are blindfolded.

How will you do it?

There was an aeroplane crash, every single person died, but two people survived. How is this possible?

I am the first thing you need while camping.
I am always with the person named Marcos Valencia and Carrick.
But Pogba Zlatan and Rooney think I am wasteful and do not need me.
I am the second to the last thing to add when you are making a patch.

What am I?