Firstly - there is a difference between 'true' and 'correct'. 'The sky is green' is a correct sentence, but it is usually not true. 'The sky is blue' is true. 'They are open in the morning' is correct English. You need the 'the'. It means that during the morning, they are open. 'They open in the morning' is also correct. Again, you need the 'the'. In this case, it refers to the time they actually open the doors. If they open the doors at 11:59 am - they open in the morning. But they are not really open in the morning! Do you understand the difference? Usually though, the two sentences have a similar meaning.

They open in the morning is correct sentence.