Is it permissible to design websites for students of the company I work for, or for anyone else, given that my primary job at the company is teaching computer fundamentals, and I might do so with encouragement from a colleague? And is the money earned from this permissible (halal) or forbidden (haram)?

The benefits of a full-time employee's time are the property of the company. Therefore, it is not permissible for him to work for himself during official working hours, especially if it harms the company. Any benefit he misses by working for others is considered the property of the company. So, if you conclude a deal with a client while working for the company, this is not permissible because it harms the company. However, outside of working hours, there is no objection to designing a website for a client or anyone else, whether they previously negotiated with the company or not, provided two conditions are met: that neither party has finalized an agreement in the negotiation, and that the designer does not know or predominantly believe that the person will use the website for illicit purposes.