I know, it can be a bit confusing at first, but that's networking. Just wait to see what you're going to be smacked with if you were to take a Cisco class though. I don't even know half of what is Cisco material :\
Basically, the way I see it, you are using your old DSL modem from back when you had Verizon as your ISP, to hook up to your Optimum Online cable service. Since your DSL no longer works, you are no longer using your DSL router as a modem for DSL, and instead are trying to use it as a cable router. Now, devices these days use what is known as DHCP, which is what allows your PC to obtain an IP address on the network as well as configure the PC for proper information such as the Gateway address you should have seen in the ipconfig command I had you run (that itself is not routable on the internet unless you have a public IP address, such as being plugged into your cable modem directly).
Now, seeing as though you have a Motorola SurdBoard modem, for what I know about Motorola modems or any cable modem really, when the modem is unable to connect to the cable network, the modem has a built-in DHCP server, which will hand out local network addresses to your PCs, so you can ultimately attach a hub or a switch to the cable modem to network your PCs together. However, if the cable modem obtains a connection to the cable network and the provider allows the modem on, the modem will switch off it's DHCP server, and the modem will go into what is known as Bridge mode, where all data such as the IP address info as well as any traffic will simply pass through the modem onto a device such as a PC or a router. If you had a switch connected to your cable modem when this happens, all but one PC would obtain an IP address and would connect, and I'm almost positive that each PC would be unable to see the PC holding the Public Internet address (the one the modem bridged to). In this case, this would possibly cause what you are seeing. However, this is not the case and you're adding a DSL router into the mix.
Now, see the problem like this. Normally when you plug your PC directly into the cable modem and the modem is in bridge mode, your PC in the case of cable networks will look for the DHCP server on your ISP's end, which in return, based on the Modem's MAC address, would authenticate your PC onto the cable network, and the DHCP server will send your PC information such as it's public IP address, DNS server addresses for looking up site names, as well as the default internet gateway that you are assigned to in order for your PC to get to the internet. This is what happens when your PC is plugged into the cable modem, and it is basically making your PC's Ethernet port act as a WAN port on devices such as a router.
However, what you are trying to do is make your DSL modem act as a router, and try to obtain a DHCP address from your cable provider, when yet the DSL router itself uses DHCP on all of it's ethernet ports, simply due to the fact that the router was meant to use the DSL connection as a WAN (wide area network) connection. Basically, you cannot just take the DSL router and expect one of it's Ethernet ports to start acting as a WAN port when no DSL connection is available. On some DSL modem/router combo units such as a Westell 327w I described earlier, though the modem can use the DSL end of the modem as a WAN port, deep in the settings of the Westell, there is an option where you can tell the Westell to take it's first Ethernet port, which is supposed to be a LAN port, and you can convert it into a WAN port in which the router can use DHCP or whatever method your ISP uses to connect to the internet through that part, and then use NAT (Network Address Translation, the thing that causes you to need to forward ports when on a router for newbies) to split up and route the traffic that comes in and out of the First Ethernet port to the other ethernet ports the router would have, as well as the wireless. The other PCs would be accepting DHCP through the modem.
Really, the only way to use a DSL modem as a router would be to set it up like I described above. Otherwise, the only thing it is really good for is acting as a switch/wireless gateway for a local network. If your cable company allows you to take more than one DHCP address from their network (which means you're paying EXTRA a month for the amount of addresses needed), you can ultimately turn off the DHCP portion of the modem and the DSL router can act as a switch/hub in a proper way. So really, the only way you can use a DSL router as a hub/switch would be as I described above, or if you had another device which would act as a router, a NAT device as well as a DHCP server (unless you manually assign your PCs static IP addresses) and a gateway to route the data.
Honestly, this is the simplest way I can tell you about what is going on before having to say everything out in a massive tl;dr post. Any questions?
If your ADSL router can't do that, might as well go pick up a cheapo router someplace and hook it up to the cable modem.