How are redirects, rewrites handled in EpiServer DXC? Is there some utility that EpiServer provides that is like bitly
I'm assuming you mean DXC Service, i.e. running Episerver as PaaS in Azure? For custom redirects you can use https://www.iis.net/learn/extensions/url-rewrite-module/url-rewrite-module-20-configuration-reference. You can also redirect domains in Episerver admin by setting up different host names and then chose a primary one that the others should redirect to.
For vanity URLs there is a property on all pages called 'Simple address'.