Praise be to Allah.
Can you take shahadah without witnesses?
In order for a person to become Muslim, it is not essential for him to declare his Islam before anyone. Islam is a matter that is between a person and his Lord, may He be blessed and exalted.
If he asks people to bear witness to his Islam so that it may be documented among his personal documents, there is nothing wrong with that, but it should be done without making that a condition of his Islam being valid.
Is marriage without a wali valid?
No marriage contract is valid without a wali (guardian) for the wife, but no non-Muslim can act as a wali for a Muslim woman. If she has no Muslim wali then the judge or the imam of the mosque or the local mufti (Islamic scholar) may act as her wali.
We ask Allah to bless you both and to join you together in goodness.
For more about the pillars of Islam, please see this answer: 13569
And Allah knows best.