Another possibility is toothpowder, which you just dip into with your wet toothbrush. The handle adaptations illustrated here can be used to modify combs and brushes as well. Extension handles are particularly useful if arm or shoulder motion is limited, although your motions will lose some force due to lack of leverage. If you use a hair dryer, mount it on the wall to leave your hands free. The bracket should swivel so the dryer can blow in any direction.
Hair washing is easiest in the shower.
However, if you bathe in a tub, a hand sprayer is useful for rinsing. Soap mitts are inexpensive and can be found at most stores where bath supplies are sold. Electric razors are easier to use and safer than blades. A bracket or clamp can be rigged to hold a razor firmly to a counter or wall while you move your face against it, or a Velcro strap holder can be used to secure the razor in your hand. A nailbrush can be mounted on the edge of the counter or sink with suction cups.