1 I enjoy dancing. (dance) 2 What do you want to do (do) tonight? 3 Goodbye! I hope to see (see) you again soon. 4 I learnt to swim (swim) when I was five years old. 5 Have you finished cleaning (clean) the kitchen? 6 I'm tired. I want to go (go) to bed. 7 Do you enjoy visiting (visit) other countries? 8 The weather was nice, so I suggested going (go) for a walk by the river. 9 Where's Bill? He promised to be (be) here on time. 10 I'm not in a hurry. I don't mind waiting (wait). 11 What have you decided to do (do)? 12 George was very angry and refused to speak (speak) to me. 13 Where's Ann? I need to ask (ask) her something. 14 I was very upset and started to cry (cry). 15 I'm trying to work (work). Please stop talking (talk).

Отвечу как участник форума по английскому ?
Все твои варианты в скобках – правильные. Разберём, почему именно такие формы используются в каждом случае.

1. I enjoy dancing. (dance)
После enjoy всегда используем -ing форму (герундий):

• enjoy doing, enjoy reading, enjoy playing и т.д.
  Поэтому: enjoy dancing, а не enjoy to dance.

2. What do you want to do (do) tonight?
После want обычно идёт to + инфинитив:

• want to go, want to see, want to do.
  Здесь всё верно: want to do.

3. Goodbye! I hope to see (see) you again soon.
Глагол hope требует to + инфинитив:

• hope to see, hope to find, hope to meet.
  Поэтому правильно: hope to see.

4. I learnt to swim (swim) when I was five years old.
После learn обычно идёт to + инфинитив:

• learn to read, learn to drive, learn to swim.
  Значит: learnt to swim – верно.

5. Have you finished cleaning (clean) the kitchen?
После finish используется -ing форма:

• finish doing, finish reading, finish cleaning.
  Так что finished cleaning — правильно.

6. I'm tired. I want to go (go) to bed.
Опять глагол want → to + инфинитив:

• want to go.
  Форма to go — верная.

7. Do you enjoy visiting (visit) other countries?
Снова enjoy + -ing:

• enjoy visiting, enjoy travelling, enjoy eating.
  Поэтому enjoy visiting.

8. The weather was nice, so I suggested going (go) for a walk by the river.
После suggest используем -ing форму или конструкцию с that, но не to do:

• suggest going, suggest having, suggest doing something.
  Здесь верно: suggested going.

9. Where's Bill? He promised to be (be) here on time.
Promise → to + инфинитив:

• promise to come, promise to help, promise to be.
  Правильно: promised to be.

10. I'm not in a hurry. I don't mind waiting (wait).
После mind всегда идёт -ing форма:

• don't mind waiting, don't mind helping, don't mind doing.
  Значит: don't mind waiting.

11. What have you decided to do (do)?
Decide → to + инфинитив:

• decide to go, decide to stay, decide to do.
  Так что: decided to do — верно.

12. George was very angry and refused to speak (speak) to me.
Refuse → to + инфинитив:

• refuse to go, refuse to help, refuse to speak.
  Форма refused to speak правильная.

13. Where's Ann? I need to ask (ask) her something.
Need (когда за ним идёт глагол) → to + инфинитив:

• need to go, need to ask, need to talk.
  Здесь: need to ask — всё ок.

14. I was very upset and started to cry (cry).
После start можно использовать и to + инфинитив, и -ing:

• started to cry = started crying.
  Оба варианта грамматически правильны. У тебя стоит to cry — это нормальный, естественный вариант.

15. I'm trying to work (work). Please stop talking (talk).
Здесь два глагола, два разных правила:

1. try → обычно to + инфинитив, когда речь о попытке что-то сделать:

   • try to work, try to learn, try to sleep.
     Поэтому: trying to work.

2. stop → -ing форма, когда прекращаем какое-то действие:

   • stop talking, stop smoking, stop working.
     Поэтому: stop talking.

Итого:
Все формы, которые ты вставил в скобках, верные и соответствуют типичным сочетаниям глаголов в английском: одни требуют to + инфинитив, другие — -ing форму после себя.