1) Is it lawful for a lady to make ablution on polished nails?
2) Is it allowed to do 'masah' on summer stockings? They are very thin and dust or water etc. can easily penetrate them.
3) What about ladies who have baby cut hair?
1) Before applying nail polish to one's nails a lady should perform ablution. All subsequent ablutions can then be performed with the nail, polish applied. However, one thing a Muslim woman must always keep in mind: growing nails is explicitly forbidden in Islam. Most women who use nail polish normally grow -- in fact "nurture" nails.
2) The real reason behind masah on socks or similar footwear is raf`ay haraj ie to provide ease. The reason is not to actually clean something. Therefore, it is okay to do masah on the stockings even if they partly stop dust, dirt and water.
3) Women are allowed to cut their hair in any decent style they want. The only haircut forbidden is that which makes them resemble men. The reason for this is that Allah has made men and women as two distinct entities upon which is built the whole edifice of a society; each has distinct functions and distinct responsibilities; copying each other in spheres specifically meant for the one is against the fitrah (human nature) made by Allah and breeds wrong attitudes, desires and inclinations.
Baby-cut hair, I believe, makes women resemble men and hence it is undesirable.