If the cover of the Mus-haf is attached to it with binding or glue, it is not permissible for anyone to touch it without Wudu. If the cover is separate from the Mus-haf like the sleeve in which the Mus-haf is placed to protect it, there is nothing wrong with touching it without Wudu.

There is a concession allowing a woman to wear a wig if all of her hair has fallen out. If you wear a wig because of that, then you may wipe over it when doing wudu. However, you must take it off when doing ghusl, because it is obligatory to make the water reach all of the skin. For more, please see the detailed answer.

There is nothing wrong with you keeping a dog for the purpose mentioned (to alert its owner that a seizure is imminent), whilst taking precautions to avoid getting the impurity of its saliva on your body, clothes and so on, as much as possible.