Kiwi IRC dev here—the network tabs not closing when you issue a /quit is a known bug right now and should be fixed soon.
It will most likely in the next release in the upcoming week, which includes closing networks via the UI also.
Your IRC traffic is never logged. Not even to kiwiirc.com servers. This ensures that no matter who asks or who looks, the IRC traffic can not be read elsewhere.
The only logging information I can see in the source code is error logs.