You have to have a router in the mix, you can not just have a modem and then have all the devices on the lan reieving internet without a router. Your modem might be a modem/router combo. If your modem has more then one ethernet port then it has a router built in.
Yes putting all of the devices behind a router would change their IP address to the router. You can configure everything so that the third set of IP numbers is the only difference (for example instead of 192.168.5.110 it would be 192.168.7.110). But as I said earlier, if your LAN devices have internet connection you must have a router somewhere already.
To do a ping you will have to do as i stated in the first post as you can not do a normal ping through the modem, through the router and filter down to the targeted computer.
I know you are trying to be able to ping computers, but what exactly is your end goal, what are you trying to find out/determine with the ping because there is probably a better solution to what you are trying to accomplish here then trying to do a ping over the internet.