Find the number

We can just use the clue about the negative of the number being the same in Fahrenheit as in Celsius which we know to be $-40$ so the number is:

$\boxed{\boxed{40}}$

We can just use the clues that the number is between 10 and 50 and its letters in English are in alphabetical order.

As the latter condition isn't true regarding eleven, twelwe, -teen, twenty(-), thirty(-) or any number bigger than forty (but <50), the only solution is forty ( $\boxed {40}$ ).

That prime number is near to Median of numbers from 1 to 68

Median of 1 - 68 is 34.5. Therefore the prime number near to 34.5 is 37.

Negative of my number when converted from Fahrenheit to Celsius is the same in both.

Let the degree of the number in Fahrenheit be $x$ , therefore the negative of that number in Fahrenheit is $-x$ .

Converting Fahrenheit to Celsius. $\dfrac{5\left(-x-32\right)}{9}=-x \\ \implies 5\left(-x-32\right)=-x\cdot \:9 \\ \implies 5\left(-x-32\right)=-x\cdot \:9 \\ \implies 4x=160\\ \implies x = 40$

Therefore the number is 40, which satisfy all the conditions above.