It is an Americn classic sit com, which is also quite popular among young people in China.   [Excellent!]

helpful to learning English    [never use "to --ing"] ---->  "helpful to learn English"  or  "helpful for learning English"

At the end of the play, she got the good job, but she stayed for Ross, luckily it is a happy ending!  [Good]

When I write these words, I can even have the pictures in my mind.  --> "As I write these words, I still have the pictures in my mind."

You could ask at www.stackoverflow.com But for this kind of question they may think you are asking them to do your homework...

"In a right triangle, the sum of the squares of the legs a and b equals the square of the hypotenuse c." Wikipedia gives a more precise version: "In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs." from http://en.wikipedia.org/wiki/Pythagorean_theorem

In the U.S. we always use 12:00 a.m. (say "twelve o'clock A M" or "midnight") and 12:00 p.m (say "twelve o'clock P M" or "noon"). Maybe some other English speaking country could use 00:00 but I've never heard of it. Normally you would write "January 2" but say "January second", etc. Sometimes people write "January 1st" for the first day of the month. "The first of January" and "January the first" sound more like poetry to me, not everyday usage.

Chair is the furniture. Seat is more abstract. Seat can also be the flat part of the chair. There are 300 chairs stored in the closet, which will provide 300 seats for 300 people. This chair is made of wood, but its seat is made of cloth. This is my seat (I'm sitting on it now), but it is your chair (you own the furniture). You should always give up your seat to an elderly person.

"Don't need to" means the same as "needn't" or "need not", but... "Don't need to" is all-purpose normal usage: "You don't need to walk, you can take the train." In the U.S., "needn't" sounds a little unusual and "need not" sounds more legal or formal: "Rule 5. Students need not attend class on Sunday."