Helen43,
The references from quietman7 are really worth reading.
When several computers are connected to a router, multicast is in the picture and pinging.Neither is a security threat.
ZA alerts because it's its job.
Kerio will too.
So will Comodo. So will any other firewall worth its beans.
That is a function of a firewall. Firewall needs advice how to handle some things.
Have you setup ZA properly - based on the alerts, I suspect not, but that's the learning process

Have you allowed the router, DNS(s), DHCP server and the local host to be trusted?
Have you alloved svchost aka generic host process to be trusted as a server in the trusted zone only?
In comodo, or kerio, have you told them what your local network address/subnet is?
Your computers ARE networked IMO. The router asks WHO's THAT? other computers answer (or don't, depending on the firewall setting), other computers ask WHO's THAT? as well?
So long as it's all local, there's just no issue. You have to permit a firewall to do its job locally between the computers you have, as well, as your ISP provider, which, too, is a private network.
How can you tell if ping tests? Read the alert carefully, quote it here if needed. Set alerts to high level, so everything is logged in \windows\ZA.log. Pings are control protocols, you should be able to see ICMP somewhere in the alert text. If you're worried, run the ShieldUp test at grc.com and see that (if?) you're all stealth and totally invisible to the external part of the internet.
Subject2: ZA free is fine. But version 7 has gotten bloated with ZA suite features which pose some conflicts for some people. It is easiest to use.
Comodo is fine, but making good firewall rules is not all that simple.
Shut down Windows firewall. It's nearly worthless.