Touching the penis with desire invalidates wudu, but touching it without desire does not. Touching the buttocks does not invalidate wudu.

In order to be on the safe side and to ensure that one has discharged one’s duty it is better to do wudu in the event of the emission of wind from a woman's front passage. If this is continual and happens in all situations, then this woman should do wudu for each prayer.

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.