Thanks for your reply
To answer the individual bits of your reply,
#1
The intent of a server is to run programs, office and game applications, from one unit, eliminating the need of installing applications and games, repetitively, on each laptop. What really comes to mind is how these public gaming server work, do the users install the game(s) on their PC or laptops before they log into these servers to play their desired game(s). I dont think so.
So why cant it be done on a smaller scale. I have a number of replies indicating subscription to and membership rates of gaming servers.
As far as office applications are concerned, it would be helpful to run it from a server, as updating will be on the server rather than all the laptops at home.
#2
There wouldnt be any heavy internet usage as I and my wife are the only internet users, at the moment. So it shouldnt be disadvantageous or problematic to run internet through the server. By the way, how does the ISP come into the equation, when all we am doing is basic internet surfing. The gaming server will not have any port forwarding for public usage. In short, the gaming server will not be accessible for outside my home for public use.
#3
This is for remote access to the server in case I am not at home. I should be able to reconfigure, if any problems, the server from my remote location, as an administrator. My family members are not computer buffs, so remote access may be required. Also, I will need to store or upload files on the server from my remote location.
#1 - That's exactly how a client/server game works. You've not mentioned what games your kids play, but lets take Minecraft as an example. You have two options when playing Minecraft, a local game, which effectively is just a single player. You'd install the game on a computer and the game will run from that computer. The other option is to run a portion of the game as a server, in Minecrafts case this typically manages things like the maps etc. However, you still need the Minecraft client installed on your laptop in order to log into the server.
So if the type of games your kids play work in this fashion, then yes you can have a client/server set up, however you'd still need the client installed on your laptops.
If you're looking at more Single-Player games, then that's not really an option for this kind of set up. There's some mentions above about running them over RDP, but frankly unless you're playing some low-end game, RDP just won't keep up.
#2 - ISP doesn't come in to play here, the "ISP Router" which you mentioned everything would be connected to is where my point was going. That piece of hardware is likely going to cause a bottleneck.
#3 - Have a look at some software called Teamviewer, it's free and is pretty much the go to application for remote support. You can transfer files from a remote location to your server over Teamviewer if you wish, but if you're going down that route, might i suggest using a cloud storage provider, something like Dropbox, OneDrive, Google Drive etc.