There are many authentic Hadiths concerning the punishment for one who drinks alcohol which say that his prayers will not be accepted for forty days. The fact that his prayers are not accepted does not mean that they are not valid, or that he should give up praying, rather it means that he will not be rewarded for them.

Mucus and spittle do not break wudu and they are pure. So, it is permissible for you to pray if mucus comes out of your mouth.

If you are reciting the Surah after Al-Fatihah from the beginning, then you should say the Basmalah (Bismillah ir-Rahman ir-Rahim) apart from Surah At-Tawbah. If you are reciting from the middle of a Surah, then it is not recommended to recite the Basmalah.