You don't have to have wudu to make du'a, but it's always better.

Wudu, the ritual washing in preparation for prayer, can be performed anywhere, but very often is performed in a bathroom. Mosques have a special area, separate from the toilets, where wudu is performed.

i dont think so but if you use the bath room it will break the wudu.

Wudu is a ritual worship to be done by Muslims; mainly before praying. It is an Arabic word that means minor ablution. Refer to related question below for more details.How do you do wudu in Islam