Routers typically don't run servers. OpenWRT and DD-Wrt based routers can provide this functionality if they have sufficient memory. This is because they are running a Linux kernel. Web server capability is done by running a web server or proxy such as Apache or Squid.
If you have several small sites to server, you may be able to use a Rasberry Pi or similar device to run your web sites. If you require more capacity, a small server running Linux can provide both firewall and Web Server capability. I started with a recycled desktop system with a second network card running Linux as both a Web Server and firewall.
I do this separation at the web server. The major web servers handle delivering content based on the Host header. The Apache web server calls this functionality VirtualHosts. It is possible to proxy some or all of a Host's traffic to a second server.
Multi-domain HTTPS is more difficult as certificates are associate with a hostname. If the domains are related, a multi-domain or wildcard certificate will work. Some servers can handle negotiating the Hostname before providing the certificate, although I have never successfully implemented this.
You will want to have your router forward HTTP requests to your web server. Most routers provide this capability by port and/or as a DMZ address.