Rise
It is common in the Bible for people to "rise up," and it would be easy to pass over the phrase as simply describing a physical action. But in fact it represents an elevation in spiritual state, moving to a more internal frame of mind closer to the Lord. Often it has to do with understanding a new or important idea; we "rise up" to a state of greater perception and enlightenment. Obviously context is crucial to the exact meaning of the phrase in a given passage -- it matters greatly who it is that is rising up, and why.
Stay
In Genesis 8:10, this signifies the period between two states. (Arcana Coelestia 880)
'To stay with,' as in Genesis 32:4, relates to the life of truth when accompanied by good, and in this instance, it means to take something in.
(Odkazy: Arcana Coelestia 4243)