If you want to use a router, especially wireless, there are 3 things you need to do.
First, change the default admin password to something unique.
Second, change the SSID to something unique. Disabling SSID broadcast does nothing, but changing it to something other than the default can help.
Third, enable encryption. WPA or WPA2 is best, but even WEP will keep the casual **** from hopping onto your lan.
If you do those three things, then you're probably ok using a wireless router. In fact, I strongly recommend that everyone with broadband use a router of some sort, wireless or not. If it's wireless, do the three steps. If not wireless, then you at least need to change the default password.
So use a router of some sort, but follow the 3 rules above and you'll be ok.
As for having both wired and wireless, yes it's possible but you really need some sort of encryption otherwise everyone nearby can hop onto your lan and use/abuse your computers including snooping for passwords, browsing your files, etc.