1. He likes jump rope.

1-1. He likes skip rope.
1-2. he likes jumping rope.
1-3, He likes skiping rope.

2. The girl is skipping in the playground.

3. Tom is jumping rope there.

4. She bought a jump rope / skip rope, and started to jump rope every day.

5. There are two jump ropes/ skip ropes on the table. Get them for me.

(Are they all grammatical? Whould you check them?)

All are fine except for one thing. In 1-3, the spelling should be "skipping" (double p to make the i sound short).

1. Out of the four options provided, "He likes jump rope" and "He likes jumping rope" are grammatically correct. The statements "He likes skip rope" and "He likes skiping rope" contain errors. "Jump rope" is the correct term to refer to the activity of jumping over a rope, so the correct phrases would be "He likes jump rope" or "He likes jumping rope."

2. The sentence "The girl is skipping in the playground" is grammatically correct. It indicates that a girl is currently engaging in the activity of jumping over a rope.

3. The sentence "Tom is jumping rope there" is grammatically correct. It states that Tom is currently performing the act of jumping over a rope in a certain location.

4. The sentence "She bought a jump rope / skip rope, and started to jump rope every day" is grammatically correct. It conveys that the subject purchased a rope used for jumping and subsequently began engaging in the activity on a daily basis.

5. The correct phrase would be "There are two jump ropes/ skip ropes on the table. Get them for me." Both "jump ropes" and "skip ropes" are grammatically correct and can be used interchangeably. They both refer to ropes used for jumping over.

In conclusion, out of the given sentences, some contain grammatical errors, while others are correctly structured.