(1) Once you have a domain name, do you also need a service to HOST the website
(2) Is there a way to insert a plugin "FotoMoto"?
(3) Can the web page which is generated by the Web module in LR, have sub-groups for different categories of photos?
(1) Yes. You will need a hosting service and plan to host your images and web pages. I recommend Hostmonster
(2) FotoMoto is supported by the legacy versions of web module plugins provided by The Turning Gate. However....
(3) This is more complex than Lightroom will allow so you may want to check into The Turning Gates new product called Backlight. It operates via the Library Module as a publish service. It supports multiple albums and even additional pages. They are planning to add a cart too which may take care of your FotoMoto needs.