Your "default gateway" setting is related to routing, and so probably really has no effect whatsoever as long as all of your devices are on the same subnet. But when you get the Internet, then the "default gateway" setting will start to matter, because devices won't be able to communicate with the Internet until they get this setting right.
Can I just hardwire the router to the Ethernet switch by an Ethernet cable.
Sure, that ought to work. Note that I'm not giving any advice related to security: things may have been separate for a reason, and you might be violating the original design/intent by connecting things up. So, I'm not saying that it's a good idea; it might be dangerous. Or, instead of just a security risk, you might introduce a switching loop that brings down the whole network (until you remove the loop). Or, doing this might be just fine. I totally don't have enough information about your network to know the full implications. But, in general, wireless access points can communicate via Ethernet, and that may help you to connect.
There certainly may be possible variations/customizations. A typical setup may look like this: You'll want a modem (DSL modem or cable modem) that gets Internet access from your ISP. Then plug your WAP (wireless access point, which you are calling a router) or switch to the cable modem.
With a very common setup, the wireless access point will want to have a default gateway that points to the IP address provided by the DSL/cable modem. Both the wireless access point and the modem can act like a "router", so the preferred "default gateway" address that other devices use will probably be the wireless access point or the modem. Which one you need to use depends on how you wire things up (like if the modem is plugged into the "Internet" or "WAN" port of a wireless access point that is acting like a router, or if the modem is plugged into another port that makes the wireless access point act like a switch... or maybe you connect the modem straight to the switch.) If you use the wrong "default gateway" setting on a device, then it won't cause any major harm; you just won't have working Internet for that device. So just try both addresses.