ping command is used to check your connectivity to the destination/machine. by traceroute you can verify the hops it goes through while moving towards ur desired destination. it will tell u the no. of hops ur destination is away from you and how much time do u take to reach the destination.
According to the scenario as shown in your attachment,
when you try to ping from 10.1.1.1, this command if ends in success will tell you that you can reach the host (that it is up, running and accessible). where as the the trace route will tell you will take the path
10.1.254.1, 10.1.254.2, 10.1.254.3, 10.1.254.4 i.e 4 hops to reach the desird host.
yes, what you say is ARP. The Broadcost will happen only at the first time, then next time it will take the entries from the MAC table,
The other thing, since you are using the switch and all the systems are working in the same VLAN, why do you want to have the IP addresses on the switches, just truncking is enough to communicate each other.
While talking about trace route, 10.1.1.2 's next hop will be 10.1.1.1. if you use trunking betn switches.
The benefit of having an IP on the switch is for management (telnet into the switch remotely) and monitoring purposes. Without an IP to it, you need to be physically at the switch and go through the console cable to configure or troubleshoot.
This document gives several answers on frequently asked questions for PFRv3 channel state behavior.
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