No, this model has no Wi-Fi capabilities. It's similar to the older models of iPod where you need to connect it to your computer via USB in order to download apps or music. The iPod Touch however does have Wi-Fi capabilities, if that is what you are after.
Not that I know of. You trade the size of the Nano for all the versatility of the Touch or Classic, which approach iPhone size. Nano is small, even if it's bigger than it was. It's almost weightless on the end of the earbuds.
It has its own built in apps, but there is no app store or place to get apps that I know of. So other than the ones it comes with, I do not believe there is a place to get new apps as of yet. That will hopefully come with time.
I haven't tried yet but I think you can't, to dowload apps you have the ipod touch. I had the ipod nano 6g and I wasn't able to dowload any apps, just reproduce music, video, pictures. This new model has nike app, if I find anything else I'll tell you