1. When my dad walked into the bathroom, the children were running around and were screaming  at the top of their voices.

The problem with this is that is sounds like they were running and screaming in the bathroom. This doesn't seem likely. As soon as my dad walked into the bathroom, the children started running and screaming (elsewhere). You don't need to repeat the second "were" - it's understood they were running and (were) screaming without it.

2. They  knocked over chairs and tables and someone had drawn funny pictures on the board.

This could be right, but you probably want to keep the tense the same. Either "They had knocked over" and "someone had drawn" or "They knocked over " and "someone drew."

3. The boy swore that he had paid for the candy bars. Okay!

4.  Yesterday, while Barbara was putting in her contact lenses, her cat had escaped. They happened at the same time. Using past perfect puts the escape before the contact lens insertion.

5. This is my last day here. I will go back to England tomorrow. ( or is it I am going back?) Either one is okay.

6  No, he went out.
 When exactly did he go out? Okay!

7. The doctor suggested bed rest for the patient. who had suffered from a bad cold. Was suffering. You want to put the suffering at the same time as the doctor's order, which was in the past. The suffering did not come before the order.

8. The national team of Brasil has only won two matches so far.Okay!

9. Geologists have now predicted that the sun will collapse in the distant future. Could be okay. You can also say "Geologists now predict that..."

10 . Are we not there yet? We have been walking for hours!  Okay!