what role can design the search page and how facets and tabs are configured?
this page about roles https://onlinehelp.coveo.com/en/cloud/coveocloudv1organizationuser_roles.htm does not describe how facets are set up and what roles can set them up.
what role can set up the search page and configure how the facets work (what facets are shown for each source and what content is included in each facet)?
If you are using Coveo for Salesforce, this is managed within the user permissions within Salesforce, not Coveo. Usually, Salesforce administrators have rights to edit the interface (which includes facets). The Coveo org is used to manage the content that will be indexed by Coveo (or view analytics data and so on).
Permissions within Coveo, do not reflect on the permissions within Salesforce, since these are two separate entities.
For information on salesforce permissions, please refer to https://help.salesforce.com/articleView?id=permissionsaboutusersaccess.htm&type=0&language=enUS&release=204.13