- وعليكم السلام ورحمة الله وبركاته
Answer to your first question, "Can I pray without wudu or tayyamum because I want to pray in secret and hide my faith..."
Ritual purity is a must for performing salah. But as to your condition, I am not really sure. You can make wudhu from the washroom, right?
https://www.islamweb.net/en/fatwa/86...n-nor-tayammum ⇐ it's comparatively a different situation but it might still help.
And to the other one, it is unnecessary to wash your private parts every time you do wudhu unless you suffer from urinary incontinence or such. This is not recommended nor advised in Islam. Plus this might result in suffering from waswas which is anything but good.
جَزاكَ اللهُ خَـيْراً