Burping does not invalidate wudu and there is nothing wrong with you burping to release gas and keep your wudu. For more, see the detailed answer.

If a person is uncertain as to whether he has nullified his Wudu, then he does not have to do Wudu until he is so certain of it that he could swear to it.

Wudu is not invalidated by seeing the `Awrah of your wife. A man’s seeing his own `Awrah or that of someone else is not one of the things that invalidate Wudu.

There is nothing wrong with praying when you have this white discharge and you do not have to change or wash your clothes because this discharge is pure.

Swearing does not break your wudu, but it is recommended for the one who commits a sin to do wudu because wudu expiates for sins, as is proven from the Prophet (peace and blessings be upon him).

Imam al-Shafi`i was of the view that vomiting does not break wudu and this is the correct view because there is no sound evidence to show that vomiting breaks wudu.

If you make Wudu, this Wudu cannot be ruled invalid unless there is certainty that what invalidates Wudu has occurred. If you are uncertain as to whether your sleep was deep or not, then your Wudu is not invalidated thereby.