I know the ICMP request contains the IP address. Is the client MAC address included in an ICMP request? What other info (if any) is included in a ping request?
An ICMP request is a layered packet which is sent over the internet. It contains the Ether layer, which has the target and source MAC address in it. It also contains the IP layer, which has the source and target IP and also a couple of flags included. And at last it contains the ICMP data. This contains a type, a subtype, then a checksum and the rest of the header, which can vary from type and subtype (E.g. The code for echo is 8 and reply is 0).
There is a lot of information in a network packet. Also note that ICMP is an part of the IPv4 protocol, so it cannot officially be carried by other protocols. Although the IP protocol can be carried on a different protocol than the ethernet protocol.
The MAC address will be changed after each passing of a router. So this is never the MAC address of the source IP address on the internet. But the IP address is definitely included and on a local network, the mac address of the client is still in the packet in the Ether layer.
Wikipedia has some readable article about ICMP packets: icmp message and different protocols with readable diagrams.
If you want to see some live requests, you can install wireshark, which will show and dissect all network traffic for you. It is a very convenient and cool tool.