There are many Linux settings that affect network. But most of them default to good values or are auto-tuning. So unless someone has been changing them, they are probably happy numbers. Here is an article about tuning the net for a high speed network:
http://fasterdata.es.net/host-tuning/linux/Be carefully, much about network tuning is counter-intuitive. For example, setting the tcp/ip packet size larger sounds like a good idea in that each packet will then have a higher data to header ratio. But, if you set it higher than the ethernet (hardware) packet size, the data will get split between two or more ethernet packets and one may get sent nearly empty. Also, because of the nagle algorithm, the system will wait longer to send a packet because it takes longer for each packet to fill.
You might be having a network problem, and your statement about ping failures whenever using your switch, indicates that it is likely. Can you swap out the switch for another and see if that changes anything?
But, on the other hand, the system monitor of your network looks very content. The same image for your CPUs does not look so happy. If I were forced to guess, I'd say you have CPU congestion -- multiple threads waiting for a smaller number to complete. This is likely caused by the design of your application and the way it is used.
I could do some analysis on your system, but I am afraid that that can be somewhat disruptive. Steve-in-Texas, has a similar Linux based setup (much faster than Windows) with a large application and quite a few users. He allowed me to send him "debug" versions of Sesame, so that I could track down problems he was having. You might want to consult with him about the process and the business implications of going that route.
If that is out of the question (in most businesses that sort of disruption is too difficult), I'd be happy to take a look at your application (with suitable non-disclosures if need be) and see if I can find where the slow-down is occurring. If it is something in Sesame, find and fixing these issues benefits us both.