Is it forbidden to apply nail polish and go out with it, knowing that it is removed for prayer, and if applied after ablution, is it permissible to pray with it as long as the ablution is valid?

If nail polish forms a layer that prevents water from reaching the skin, it must be removed when performing ablution (wudu'). If it is applied after ablution, there is no harm in that, as merely applying it does not invalidate ablution.

As for wearing it outside the home, there is no harm in doing so if the hands are covered with gloves or similar coverings. However, if the hands are uncovered, it is not permissible, as it involves displaying adornment to non-mahram men. Allah the Almighty says: "And let them not display their adornment except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers, their brothers' sons, their sisters' sons, their women, that which their right hands possess, or those male attendants having no sexual desire, or children who are not yet aware of the private aspects of women. And let them not stamp their feet to make known what they conceal of their adornment. And turn to Allah all of you, O believers, that you may succeed." (An-Nur: 31).