X-23's mutant genes are derived from Wolverine. Because of this, she shares much of his innate healing ability. Logan's mutant gene was specifically isolated to be reproduced in X-23 to exhibit the same characteristics, albeit with differences pertaining to her gender and the mild random effects of claw distribution based on conception as well as environmental needs due to the cause of her powers awakening. The process and manner used caused her claws to manifest in the best manner to suit the duress she was under as a natural response to combat it. Where Wolverine's claws were manifested first initially under a combat/anger scenario, X-23's first manifested in both offense and defense manners that allow her to use both her arms and legs as tools to protect herself.
Her inherited healing factor though is the point of discussion for today!
Rate of growth for her hair:
Her hair should grow at a rate that corresponds to the amount of cellular regeneration she undergoes. Essentially her healing factor regenerates her cell structure at an accelerated rate of a normal person with no decay that would be associated to the aging process. So in theory, her hair should grow at the same accelerated rate while her ability is active.
It's never been shown to grow beyond the set length she keeps. Though there are two possibilities with how her powers work. Either her hair is perpetually growing at an accelerated rate, or it only grows at such speeds when her powers are directly active to do so. As such, when her healing factor is not in use, her hair may not be effected by it. She could in theory get a hair cut, but the first major trauma she suffers make it jump several inches in length that corresponds to the natural human hair growth rate of the time a person would need to recover from such an injury.
When exposed to excess amounts of radiation, her body may compensate from the potential effects, but until the radiation's half-life expires, hair growth may be minimal. Once the effects do wear off her eye brows and eye lashes should grow to normal length first because of the protection roles they play in instinctual reflex reactions that keep dirt, sweat, and other debris out of the eyes.
Damage to her brain is dependent on a host of factors of whether or not tissue itself has been removed or altered. In cases where tissue may be incapacitated, in theory it could regenerate back to where it belongs without any memory loss. Such things as high levels of radiation exposure may temporarily incapacitate it, but so long as the tissue itself is intact albeit inert until her powers regenerate it, her memories will be unreachable until said biological function completes. In such actions as brain matter being directly lost where new tissue must be grown, memories could be lost. It's dependent upon the manner of damage taken, and how her power regenerates said tissue. Dead tissue that's revived has a different effect than tissue that needs to be fully regrown per the biological functions associated.
Full Limb Regrowth:
X-23's healing factor activates upon cellular damage. Anything that compromises her natural state, such as limb loss, is regenerated at an accelerated rate. We've seen this previously in X-Force v3 with her having to grow her entire arm back and have her claws re-implanted in that arm to replace the newly grown bone claws. Unlike Wolverine, her healing factor is not compromised or hindered by constant regeneration from excessive amounts of adamantium poisoning nor is it hindered by the fact said type of coating would block all anti-body production. The lack of adamantium coating her bones means she would have a potentially exponential rate of healing acceleration compared to what Wolverine does because of the biological processes of bone marrow still being active in her body as opposed to how his is blocked by the metal.
Her healing factor works at a rate of regeneration on a cellular level. This means there is no cell decay that would correspond with the aging process. Once those cells reach peak maturation, they are continually rejuvenated into staying in peak condition from all natural decay. This keeps her outward appearance from aging beyond her prime maturation point. What that point is though has yet to be fully seen. Usually it can range in normal physiology anywhere from 25 to 35. Her healing factor, barring hindrances, would essentially stop aging at her physiological peak, but may also slow the climb to it as well.
When dealing with such characters as X-23, to maintain what is the natural essence of her abilities, you have to look into how that ability corresponds to her physiology and history. This is something that a writer must do with any character's attributes they are handling.
For further references on how her healing factor has been used in comics. See NYX v1 issue 7, X-23 v1 issues 1-6, Avengers Arena issues 1 and 18, X-Force v3 issues 17-21, All-New X-Men issue 19-20, and New X-Men: Academy X issues 30-31.