1) You may be interested in Catalan number.
"$C_n$ is the number of standard Young tableaux whose diagram is a 2-by-n rectangle. In other words, it is the number of ways the numbers 1, 2, ..., 2n can be arranged in a 2-by-n rectangle so that each row and each column is increasing. As such, the formula can be derived as a special case of the hook-length formula."
2) All you need is to separate $2N$ numbers into two sets of $N$ numbers. After separating them, you have only one way to arrange the $N$ number of each set.
1) Forget Catalan number. Your answer tells us that your question is saying nothing about the arrangement in each colum. Now we can get your answer in the following way :
Imagine that you take two numbers as a pair, and take another pair, and so on. You can get the arrangement of yours in this way. Notice that each pair has only one way to arrange them.
2) The answer is
because this says that you take $N$ numbers from $2N$ numbers. Once you take $N$ numbers, this set of numbers has only one way to arrange from smaller to bigger. Then, we know the answer is this.