I'm not a native speaker, but I'll have a go for the fun of it:
I would use 'that' with these ones:
5 a) He warned me the roads were dangerous.
8 a) He suspected John was a thief.
9 a) He promised he wouldn’t be long.
11 a) He complained the soup was cold.
12 a) He explained the waiter had a lot of tables to serve.
13 a) He suggested the restaurant should employ more waiters.
14 a) He wrote he wasn’t satisfied with the service.