The answer to your question will very much depend on the specifics, but I will try to give some rules of thumbs.
Do I need to mention school A as one of my affiliations?
This depends to a large extent to the relevance of you being affiliated to A to your publication. If the subject of your publication and/or the venue where you want to publish are even partially related to A's activities, then: yes you should present the affiliation.
Let me give an example: suppose you work as accountant at an oil company, and in your spare time you have done some environmental related research. While you might consider you to be unbiased, and the daily activities of your work do not have direct environmental aspects, it is still important for the readership of your article to know that you are professionally affiliated to an oil company. They can then decide for themselves whether they want take that into account when reading your article.
Do you think school approval (review) at A is usually required ?
This depends very much on the rules that A has in place for their employees, and on your own agreements with A. Read your contract, any other agreements and regulations and ask your boss (or any other relevant person who has the authority to make decisions about this).
Some companies have strict regulations about publishing. You can try to make deviating arrangements, but if you do, make sure they are clear and they are in writing.