Would it be legal under trademark and copyright laws to name my search engine company "Not Google"?

From what you ask, it appears that you are asking if you can use "Not Google" as an identifier of your goods or services. This is a trademark use. Under trademark law, you can only use someone else's trademark as a "fair use" if you are using those words descriptively, for example the words are otherwise used to describe common objects. For example, you can use the word "apple" to describe the fruit and the company cannot stop you from using it in that way. In some courts, there is also "nominative" fair use where you can use someone else's trademark to refer to the actual trademark owner or its goods/services. It doesn't sound as if you are doing either of those here. Copyright does not seem to be implicated from what you have asked, however if you decide to pursue this further I strongly suggest you consult an intellectual property attorney.