No, university symbols or logos such as Jayhawks and KU are protected under federal unfair-competition laws. A trademark protects words, names, symbols, or devices that identify a specific product for consumers. You may not use them to make sales without a license or permission.
If you try you may be sued and will lose.
The use of a college or university name for commercial use purposes is subject to right to publicity laws in each state. Some states prohibit the commercial use of names including college or university names, while others do not. In addition, many colleges and universities implemented policies prohibiting commercial use of their names unless specifically licensed or authorized. You need consult with a local attorney specializing in such matters.
Disclaimer: This answer should not be relied upon as legal advice. I am admitted in the States of New Jersey and New York and before the United States Patent and Trademark Office only. The response above does not form an attorney-client relationship, nor it is intended to be anything other than the educated opinion of the author. Consult an attorney in your geographic area before you act on any of this advice.
Of course not. Schools protect their IP rights very aggressively, including their trademarks on their school name and logo and mascot and "trade dress" (the way the school's products look in terms of color, font, shape, silhouette, etc.). They make retail products, or license others to do so, to various clothing, toy, calendar, dishware, etc. etc. vendors, and any unlicensed use would very likely get you sued.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Absolutely not. But colleges and universities usually have licensing departments who provide licenses in exchange for reasonable royalty payments, If you want to use a college or university name on a retail product, an IP attorney can negotiate the terms and provisions of the license agreements that you need. Of course, a college or university has no obligation to grant you a license. But most will happily do so as long as your retail product does not adversely impact their image or reputation.
You should not feel bad that you are confused on this issue---I am always surprised about how even sophisticated business persons misunderstand and underestimate the importance of intellectual property rights. I believe that essentially every small business person should have an IP lawyer on retainer---IP issues arise regularly no matter what business you are in, and no matter where you operate such a business. One of the common reasons why businesses fail financially is failure to adequately account for IP issues (patents, copyrights, trademarks, rights of publicity, and others). You have to worry not only about infringing rights of others, but protecting your company's IP so that your competitors cannot take advantage of you. The question you ask is common--and many people go into business based on the misconception that they somehow are free to use IP rights belonging to others--quite obviously this can be a fatal mistake. Law suits are expensive, and the last thing you want is to find yourself in a federal court with a bunch of lawyers fighting among themselves about how much you owe the plaintiffs, including how much you should pay in legal fees. The amount you pay for competent legal advice now will result in huge savings in the long run.