What is both possible and impossible at the same time?

Begin with a word, five letters to my name,
Remove the first and last but I am the same
Take out my middle and still, I remain.
What word am I?

I was speeding and ran through a stop sign, yet two traffic officers who were there do nothing about it.

why?

Find out the missing number in the picture attached:

Some friends went on a vacation to a resort. It was raining heavily and it kept raining for thirteen days.

When it rained in the morning, the afternoon was beautiful and when it rained in the afternoon, the day was blessed with a clear morning.

Overall, the friends experienced eleven nice mornings and twelve nice afternoons. Can you find out the number of days they spent on vacation?

What can you catch but never throw?

How many bananas can you eat if your stomach is empty?

On a certain day, John celebrated his birthday. Two days later, his older twin brother Jacob celebrated his birthday.

Is this even possible? How can it be?

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

What can run but never walk, have a mouth that never speaks, have a head that never weeps, and have a bed but never sleeps?