N access point/router is an overkill for home applications unless internal home network has several (~5+) computers, which exchange a lot of data. Speed of a G WiFi network is already much higher than a typical cable or DSL internet link so upgrading to N WiFi is pointless if Internet speed access is the only consideration.
The difference between an N and a G router is all in what the purpose of your router is for. You need to see if you are using it for a personal or business use.
You can use router of G or N type and the same type wireless adapter. Of course you can use wireless N adapter with G-type router and reverse.
N adapter should work with G router but you probably won't gain any speed. Both ends need to be N, I think -- and preferably from the same brand.
The same as with wireless G card and G router.
Yes, however, your internet speed will only be that of the G router which is comparably slower than that of an N router.
You would need a 802.11 G.
No.
You can't and you don't want to. A "N" router will work with "G" network components and will additionally support "N" hardware at a higher speed. Converting it to "G" would be a step down.
N is better. It is faster.
Usually they are, you have configure the router for G wireless networks.
In the wireless properties choose "compatibility mode for G" or something like that. If you do so your router will allow both types of wireless cards G and N.
Your wireless adapter probably worked better with the old router. Or the new router is a lower end N router and the old one was a high end G