My car MOT runs out in a month, if it is retested before expiry and fails is previous cert stil valid?

Yes. the think to do is (if it is not the car's first MOT) is get it tested 14 days before it is due and you wil have time to have it repaired before the original runs out. if it pes you will get one year and 14 days before yor MOt runs out because they will date it's validity to start from the date your old one expires