As you can start a PC through its 'WAKE ON LAN' feature on private networks, can you trace someone's online status through their MAC address over the internet ?
The MAC address is link-local. When a machine emits an ethernet frame, that frame is tagged with the MAC addresses of the source and destination hosts, which are both connected to the same local link (plugged to the same LAN, associated with the same WiFi...). Normally, (at least) one of the two machines is a router: the frame contents are an IP packet, and the router will put that IP packet in a new frame on another link, and send it on its way. Only the IP packet thus travels; the original frame header, including the MAC addresses, remains local.
Thus, the MAC address of a machine cannot be viewed from other machines which are not part of the same LAN. It cannot be seen, and therefore cannot be used for tracking, from remote Internet places. Similarly, you cannot send a packet meant for a specific MAC address if you are not already on the same LAN. Global routing works on IP addresses, not MAC addresses.
Now there are exceptions to all rules. IPv6 is the next version of the Internet protocol, waiting to be widely adopted (I am not sure it will ever happen, but hey, one can only hope). The relationship between ethernet frames and IPv6 packets is still as I explained above; however, IPv6 may imply a leak of MAC addresses.
Indeed, most user systems nowadays (as opposed to servers and routers) obtain a dynamic IP address when they connect their LAN or WiFi access point. With IPv4 DHCP is used, and, usually, the user's machine is allocated a "private" IP address, that will be converted with NAT when external systems must be contacted. With IPv6, the IP address space is much enlarged, so that NAT is no longer necessary, so the machine's IP address will be visible from the outside; furthermore, IPv6 dynamic address attribution is done with NDP, and (pay attention, that's the important point here) the machine's dynamic IP address will include, as a subfield, its MAC address (the MAC 48 bits are reused as part of the 128-bit IPv6 address).
A consequence is that, from an IPv6 server's point of view, client machines can be tracked. Not that the server would somehow know if a given machine is up and running; but in the sense that the server may notice when two client connections, from distinct IP address, are really from the same hardware (e.g. a laptop, the user having just moved from his office to his home). This is not the kind of tracking which would allow a "ping" request (the tracking server must wait for the system to connect to it), but it is a privacy concern nonetheless.
Extensions to the protocol have been defined, to generate IPv6 addresses differently, in a less trackable way. Most modern desktop/laptop OS implement these extensions, so the tracking explained above does no longer work, or will cease to work in the near-term future (and nobody uses IPv6, anyway).
No, MAC Addresses are not routed across the internet.
There is no way to get the MAC address of a computer that is not on the same subnet without some sort of software installed on the computer.
If you are wanting to know the status of a device on the public internet, you should be checking their IP address instead.