You definitely want to fix this.
Your Belkin router should be able to operate in two modes - as a bridge and as a NAT. It sounds like right now it's just bridging your ISP's connection so you are, indeed, seeing a swath of other customers in the area. By extension, they can also see you, which is why you want to fix this.
Check the router's configuration and put it in NAT mode. You should be able to set it up to get a routable IP address from your ISP as well as use a separate private class network (e.g. 192.168.x.x, or 10.x.x.x) for your private network that includes your Mac and any other devices you have.
NAT shouldn't have a significant impact on performance, but it does elevate the security level.
Follow Camelot's advice.
In addition, some routers control NAT services via whether or not you enable the DHCP server in the router. You should have it enabled. Some routers actually have a separate control for NAT server, if yours does it should be enabled.
With respect to your Mac looking for other computers with shared services, it does not work that way. Rather the other computers broadcast their availability, and your Mac sees that. You Mac is not polling, it is just listening.
And whether or not you are broadcasting availability of services on your Mac is controlled by System Preferences -> Sharing. If all are disabled, then you are not broadcasting anything.
Regardless, you need to figure out why you are seeing other services if you do not have any in your own home. So follow Camelot's advice.
Our internet is working so much faster than it used to with the old Base Station, that I hesitate messing with any settings!
Besides that, all that stuff about NAT is totally over my head. I can tell you that when I go to Network in System Prefs and click Advanced, then TCP/IP it says it is using DHCP on the line Configure IPv4. It seems to me that with the Base Station the address started with 10, and now it starts with 192 with the Belkin.