can it work over a wireless home network? or does it require an ethernet cable?
Is there any TIVO device that will work on a wireless basis to do what the Mini does?
No the Mini is not wifi. If you don't have a ethernet connection, you can use something called MoCa connection which works over your coaxial cable that's what I use and it works just fine. The TiVo Roamio Plus and Pro are able create the network.