IIS is available on Windows - you need to turn it on under programs and features.
I wonder if a web server being accessed via family members only would violate the FIOS TOS which is really aimed a high bandwidth usage.
To do what you want to do you need to purchase your domain name from some provider, netsol, godaddy register etc.
They'll publish the domain to dns for you. They'll also typically provide you for free with your domain name a one page web site or forwarding for you domain name to the address and port of your choice.. To accomplish that on a fios dhcp assigned router wan address you will need an account with somebody like DynDns which will track your current ip address to the account.
So basically that means you would buy your domain name e.g www.myhome.com
get a dyndns account e.g myhome.homedns.org
build your web pages using a tool of your choice and publish them via the web server on a machine with a static ip address on your home network, also configure the web service listening port/s
configure the router to update your dyndns account
configure your router to port forward all traffic for the listening port on your web server to the static ip address you defined for the web server machine
configure the isp account to forward all traffic for www.myhome.com to myhome.homedns.org:xxxx where xxxx is the listening port on the web server.