>>Is it possible to host a static website on any of the Linux Web Hosting packages?
Yeah, you can host a static website using the linux hosting packages provided by Afrihost.
Refer the URL : https://www.afrihost.com/site/product/linux_hosting
for package details.
>>Additionally, I understand I would not have access to the web server's settings when using one of those packages - is that correct? I am particularly interested in URL redirection.
Your website will be hosted in a shared server and there are many domains hosted in this server. Hence Afrihost cannot provide access to server settings. However the support team will assist you with domain related changes. You will be provided with a control panel to manage the domain settings and can set a redirection using the control panel.