If a hard-wired computer or peripheral has a 10/100 Ethernet card, does that hurt overall network performance?
No, but the communication between this type of device and the rest of the network can only be at the highest bandwidth available. That is these devices can only achieve a maximum potential bandwidth of 100 Mbps even if they are connected to a 1 or 10 Gigabit Ethernet network.
Will two Airport Extremes help improve network wireless performance?
Depends on what you mean by performance. With two or more base stations, you should be able to extend the current wireless network so wireless clients that were farther away from the existing base station would perceive "better" performance as they would have a stronger signal to work with ... but extending a network does come at a performance cost due to the bandwidth overhead required to operate this type of network. The older Wireless Distribution System (WDS) method of extending networks could use up to half of the available bandwidth. The newer 802.11n version of extending a network has a considerably less impact penalty.
Will switching from a 4th generation to a 5th generation AE help?
Potentially yes. Please check out this excellent SmallNetBuilder article where they compare the 4th & 5th generation Extremes to get an idea of what you can expect.