What is the purpose of each of the extractions (washings) the crude product? in other words, what is removed upon washing with (a) sodium hydroxide, (b) hydrochloric acid, (c) water, and finally (d) brine?

Generally, the purpose of each extractions of the crude product is to purify the product, take out the less useful products and obtain the more useful ones. Each of the washing solutions mentioned ( (a) sodium hydroxide, (b) hydrochloric acid, (c) water (d) brine) would be aimed at taking out a specific substance that is not the desired product. However, it would be difficult to specify what would these be unwanted products are since the whole experiment is not specified and even the reaction involved. I could not really know what are the substances that are dissolved to those solutions used for washing.

Maybe you should get some from your teacher and learn more so you can get your grades up and maybe study a bit more