老实 is to describe that a person is very kindly. He never sets up someone. He never breaks any rule and law. He always believes other people. When someone is too 老实, he can't distinguish any lie. In this case, 老实 is a bad word, meaning someone is fool because he always believes other people. It seems that he has no brain.
诚实 means "no lie". 他是个诚实的孩子 means he is a child never lying.
忠实 means faithful. For example, 我是巴西队忠实的球迷(faithful fan). 这名法官忠实地执行了宪法。 This judge follow out the constitution faithfully.
诚恳 means a kind of believable feeling. When someone does something or says something that makes you feel that he doesn't lie, you can use 诚恳. For example, 他诚恳地向我道歉. He says sorry to me and I can feel that he really wants to say sorry to me.
真挚 is about emotion, such as 友情、爱情. It means that the emotion is true. For example, 他们一起经历了各种磨难,他们的爱情一定是真挚的。