Are these statements correct?
Madra a bhí ann. - It was a dog.
Bhí cat ann. - There was a cat.
Bhí sé De Luain. It was Monday.
Bhí sé uafásach. It was terrible.

Yes, except for the Monday one. It might depend on the context. If it's something like "it was Monday, the sun was shining and I had just made a cup of tea..." > 'An Luan a bhí ann' or 'Bhí an Luan ann'. I don't think you could ever say 'bhí sé De Luain'.

Thank you very much.