The Actiontec rev I (the "advanced" router) runs at 1Gbps. Cat5e and Cat6 will both work the same with this router. Cat6 provides an upgrade path if you think a 10Gbps Ethernet network is in your future. It provides no difference in performance or operation at 1Gbps. For most people, Cat5e is fine.
802.11n runs faster than 802.11g (130Mbps vs. 54Mbps), however in my experience the range is lower with the Actiontec rev I. It may work better for you, but there is no way to tell in advance due to all the variables the effect WiFi performance. A wired connection will give you the best performance.
Note that a 100Mbps Ethernet connection is faster than most speed tiers on FiOS. Unless you have a service above 100Mbps, you don't need a 1Gbps Ethernet to get all the speed out of your internet connection. 1Gbps is helpful if you are going to be sharing lots of data between devices in your home, but does little for the link to the internet. In short, if you're concerned about internet connection speeds, the old 100Mbps router is quite fast when using wired Ethernet.
Have Fun.