All computers must have an address, much like a house or residence must have an address. No two computers can have the same address, and in your home, each computer has their own, private address. However, when connecting to the Internet, the computer will use the address of the router, which is a public address. The router allows multiple computers and devices to connect to the Internet using one public address. If it did not work this way, there would not be enough addresses available for everyone to be on the Internet at once!
As far as adding a server to a home office, it would depend on what kind of server you would like to setup. If you are looking to have computers connected to the Internet through the server, you would likely still need a router. If you use multiple computers in your family, a server at home can be beneficial. It could be used to perform backups, to store all your pictures, videos, and media in one central location. All computers would then have access to all files. It could be set up as a print server to allow one printer to be used by multiple computers. There are other uses, as well.
One thing to remember about a server is that it must always be on to be effective. However, it does not have to be a current, up to date computer. It can have older hardware and does not have to be a fast computer, depending on the use. It could even be set up without an Internet connection, if you are using it for a file server or for backing up your systems. All it would take is some knowledge in how to set up a home network.