They (play) football, they're going to watch it.
- This sentence is already correct. It means they are going to watch a football game.
He (walk) to school, he's going to cycle.
- This sentence is also correct. It means he is going to ride his bike to school.
I (have) lunch because I'm not hungry.
- The correct form here is negative: I'm not going to have lunch because I'm not hungry.
She (go) to bed early because she's tired.
- Correct: She's going to go to bed early because she's tired. (Or, more simply: She's going to bed early...)
I'm interested in athletics, so I (watch) the Olympic Games on TV.
- Correct: I'm going to watch the Olympic Games on TV.
We (go) shopping, we're going to stay at home.
- This sentence is a bit confusing. It seems contradictory. If they're going to stay at home, they're not going shopping. Perhaps you meant to say: We're not going to go shopping, we're going to stay at home.
I (finish) my homework tonight, I'm going to do it tomorrow.
- This sentence is also contradictory. If you're going to do it tomorrow, you're not going to finish it tonight. Correct form: I'm not going to finish my homework tonight, I'm going to do it tomorrow.
I (write) a long email and then send it to my friend in Spain.
- Correct: I'm going to write a long email and then send it to my friend in Spain.
We (buy) a new car because our old car is fine.
- If your old car is fine, you probably won't buy a new one. Correct: We're not going to buy a new car because our old car is fine.