Is an inner stillness of mind necessary to spirituality?
Yes, imo. Without it we really are flapping about like a fish on land and our chances of getting back to some water are entirely dependent upon someone or something picking us and dropping us in it.
If so, must we always have an inner stillness of mind for genuine spiritual awareness, or is stillness more like an occassional experience that enriches and informs our spirituality?
Its both in a way. Not possible to stay in stillness indefinately but there are varying degrees of inner-quiescence that can be maintained while active. There has to be a movement towards stillness at least. If I wasn't spending 99% of my time in denial and willful distraction I'd be working to stay in a more or less permanent state of meditation.
Can we have a genuine spiritual awareness if our mind flaps about like a fish on land?
Maybe a certain awareness can persist, but to grow we need to become quieter.
I like this theory of harmonising yin and yang which suggests this has to be the case: The forcefulness of yang must yield to yin. Yang, defiant & excitable by nature, is yet only fulfilled when it gives over to the governance of yin. The receptivity of yin should certainly lead yang. Yin, submissive & reposeful by nature, is yet only fulfilled when it is given the governance of yang.
Is spirituality grounded in something deeper or more primal than mere day to day consciousness?
Something more basic, more fundmental than day to day consciousness? Yes I think so. Its so often said that God is found within, that we already have Buddhamind, that the Tao must be forgotten to be followed, etc. Its like we get stuck in confused patterns of thought and behaviour, become so familiar with them that we fail to realise how we're stuck, and need to withdraw from them in order to recover what was potentially always available to us - our direct line to Spirit.