Our product and our sum always give the same answer. Who are we?

I am eight letters long - "12345678"
My 1234 is an atmospheric condition.
My 34567 supports a plant.
My 4567 is too appropriate.
My 45 is a friendly thank-you.
My 678 is a man's name.

What word am I?

A man has to get a fox, a chicken, and a sack of corn across a river.

He has a rowboat, and it can only carry him and one other thing.
If the fox and the chicken are left together, the fox will eat the chicken.
If the chicken and the corn are left together, the chicken will eat the corn.

How does the man do it?

I have one of the three numbers: 1, 2, or 3 in my mind. I speak only truth. You can ask me just one question for which I will only reply in yes or no or don't know. What question will you ask from me so that you are able to know the number?

A girl has as many brothers as sisters, but each brother has only half as many brothers as sisters. How many brothers and sisters are there in the family?

A man calls his dog from the opposite side of a river. The dog crosses the river without a bridge or a boat and manages to not get wet. How is this possible?

The more it dries, the wetter it becomes. Can you tell what it is?

There’s a girl who has a large family. She has an equal amount of brothers and sisters, but each brother only has half as many brothers and sisters. What’s the correct amount of brothers and sisters?

A man lives on the fifteenth floor of an apartment building. Every morning he takes the elevator down to the lobby and leaves the building. In the evening, he gets into the elevator, and, if there is someone else in the elevator, or if it was raining that day, he goes back to his floor directly. Otherwise, he goes to the tenth floor and walks up five flights of stairs to his apartment. Can you explain why he does this?

Solve the equation in the image by looking at the pattern closely.