A cheap wireless router can cause lots of problems. If the wireless cuts in and out then it can cause wired connections to fail also. It does not make sense but the more things you connect to a single ip, the more things can go wrong.
If you connected a router keep in mind that a router is designed to let things pass through in one direction but not the other unless it is specifically configured.
I had a cheap wireless router a few years back from AT&T and my streaming video from HULU kept losing the connection. They upgraded to digital and seemed to fix a lot of DSL problems.
Not really. That isn't wireless cutting in and out, that is the ROUTER cutting in and out. I had a wifi router that went on the fritz. When it was dying it wouldn't route traffic over the WAN port, nor could you access the management page. The wireless and LAN ports all worked perfectly though, so I switched to using it as an AP (routing/management would die after 10-14hrs of use as a router). I've also had a router that would work perfectly, but the wireless AP in the router would occasionally conk out and the whole thing would need to be rebooted for the wireless to work again. Still routed traffic over the WAN port just fine though.
That is completely not true about being designed to only pass traffic one way. It does ROUTING, which means moving packets both directions. It functions in effect as a firewall, limiting inbound traffic only to packets that were requested from inside the network, unless you do port forwarding. This is because of Network Address Translation. If something inside doesn't request something from outside, then the router doesn't know where to send a packet when its coming from the outside (without port forwarding).
Otherwise its an "unaddressed packet" as far as the router is concerned and it drops it.
This isn't an issue in 99.9% of use cases, because for a personal network, you generally only want data coming in from the outside when you are requesting it/connecting to a service.
That is the basics, I can get more depth in to how a router works if you'd like.
As for a cheap router, well, yeah, sometimes they are bunk. However, pretty much any cheap router with gigabit ports on it should be able to handle routing a 100/100 internet connection at full speed. The really cheap ones might not be able to handle much more than that, but they'll sure handle a 100/100 connection fine.
Its once you start talking about 200+Mbps internet connections that low end gigabit routers might not be able to keep up. Then again, there are some high end routers that won't keep up with that either, depending on the services running and the type of internet connection you have going on.
A $40 gigabit router should handle your 100/100 connection fine. My personal suggestion would be a TP-Link WDR3600. Nice and solid, good wireless, very good routing performance. ~$50 new. I've bought two of them open box for $35 each.
Also I cringe that you were or are running your desktop straight to the internet. Software firewalls are only so good. Granted routers have plenty of security vulnerabilities too, but at least you'd have to get through an exploit in the router before you could ever try to target something behind the router. Direct connected its only software firewall and a prayer protecting you.