Basic information aside, her age is unknown. Same with her true name. Despite her energetic attitude, she doesn’t ignore the toll of her age. Being easily hundreds of years old, and seeing the amount of death and destruction as she has, she can often be very sensitive and grumpy when reminded of who she is. She often gets this way when her companion urges her to talk about her past. Her philosophy is that you can never really trust someone. Even though she is close with Ruby, Annie refuses to discuss her past. At least, not yet.
As for her name, and choice of disguise for her TARDIS, an Anesthetist is one who gives another anesthesia. Although anesthesia puts people to sleep, it is still a medicine. Therefore, she is one who strives to deliver help as much as she can. This ties in to her choice of TARDIS disguise. A wishing well is used to wish for things. She wants to be able to make wishes come true.