With regards to employees assigning to admins and the ticket being marked as closed - this was something we faced here, but it's pretty easy to set the status of the ticket under the OPTIONS area.
Rather than our employees assigning to other employees they are only allowed to assign to themselves if they're working on something specific (or a scheduled site move). Once the task is completed, that flag is removed, so that if a customer responds all users see the ticket again.
We also created a seperate user for ADMINS, and ALL users are allowed to see tickets assigned to other employees. This allows staff to make sure that other staff are on the ball (or can assist as needed).
With regards to tickets closing without replies - that's just poor management. Remove the AUTO close setting in the automation section - or set it to close after 150 hours of no replies. Honestly if an issue is not resolved in 150 hours, you've got bigger problems than a ticket closing - you've got people not doing their job!
I can understand that the ticket gets set to ANSWERED - you just have to teach your staff how to reply to such things. If they don't adhere to the rules, you write them up and then later terminate should the issue become pressing. I've never had an employee NOT understand processses and procedures.
The other option is to tell those staff members NOT to answer the ticket. if they can't answer it, don't touch it - let someone else do it. Then use the Rules Escallation process and use that to notify admins or senior staff.
Kayako is nice, but it had ZERO integration with WHMCS for a long time - too long. Sure it works now, but what about the 1.5 years where it didn't work!
To each their own really - I much prefer keeping things within WHMCS - ease of use and easy to track. I stil have my licenses here for Kayako - just don't use them.