Hi LeonardB,
Here's a doc that HP released some time ago relating to pinging switches...
Inconsistent Ping Response
Basically, if you want to use ping to check if a switch is up or down then that is fine. As the documet states, pinging the switch will have a lower priority to respond than traffic passing through the switch - this is to stop ping of death attacks against the switch. It is always advisable to ping an end device to collect latency response times, as switched traffic takes higher priorityover management traffic, but for just seeing if a switch is up or down pinging it is fine, unless, as your network people state, ping responses on the switch are disabled. If your switch is dropping pings then you either have a problem with the switch itself or the CPU load is so high that the pings to the switch are dropped because it can't respond in a timely fashion.
To answer your questions...
1. No not at all, the amount of pings you are talking about is insignificant to the amount of packets the switch will be switching in a real world scenario.
2. No, not that I am aware of.. Ping is the first basic method used to probe any device on the network so to say they should never be pinged is incorrect.
3. Without being able to configure any other manageent protocls on the swithc (SNMP/logging to NMS,Syslogger/Etc) you are somewhat limited. Maybe some other guys here have some ideas, but I cannot think of any.
The only thing I would say is that pinging the management IP of a switch does not give you the view that traffic is being switched correctly. To monitor that you would need to ping the end devices attached to the ports on that switch. Pinging the management IP will just confirm that the management IP is responding, or subnet gateway if it is used as a L3 router, and the physical switch is online - or subnet gateway if it is used as a L3 router.
HTH.