We all know that ping is the most common way to establish layer 3 connectivity. Layer 1 connectivity is established by link light. Layer 2 connectivity is establish by checking if link protocol shows as up (or you see the MAC address), but how about non-ethernet layer 2 connectivity? How about layers 4, 5, 6, and 7?
So, the question is: What are common ways to unequivocally establish connectivity at layers 2, 4, 5, 6, and 7?
Some ideas that pop into my head:
1) telnet to port 80/443 for layer 7 connectivity
2) use netcat (how?)
Layer 2 requires layer 1 and layer 3 requires layer 2. Everything above that requires a functioning transport layer (3) and therefore tests above that are application specific and will vary between what type of service you are running.
Like you have already said, telnet which is an application so layer 7 is a handy way of testing socket connectivity between devices. It also requires all layers below it to be working and therefore I would call it a complete test of all the layers involved to have a working IP network.
If you are debugging connectivity(Layer 3) you need ping and traceroute.
If you want to check open ports you need tcptraceroute(Layer 4 TCP) or nmap (Layer 4 TCP/UDP), but it's pretty hard to debug UDP.
openssl s_client -connect will help you to debug SSL(Layer 5).
For everything else use tcpdump/tshark.