How do I develop a live support chat application?

When beginning a new project, you probably shouldn't concentrate on _how_ another person has done something, but only on the _what_ has been done. You appear to have a reasonable handle on the _what_. _How_ such systems have been done should never enter your thoughts; their way may be awful. _How_ would _you_ like to do it? What tools do _you_ have that can be applied to the challenge?

I agree with using WCF or similar. I do not agree with "the server/service will then call all connected customer support agents until one answers" (I think "push" is a very poor model).
Personally, I'd use a database to track and contain all these conversations.
Agents start their clients, log into the system, and bring up a list of conversations -- perhaps polling the database every five seconds or so.
When a customer starts a conversation (hopefully with a clear product-related question), it gets entered in the database and it shows up on the agents' lists -- "Joe in New York asks: 'Does product X support XML?'" .
An available agent can then select the new conversation and off they go.

That sounds like the basic features required, but you could add more once those are functioning.
The list of conversations can have other features, like color-coding for conversations that have been waiting more than some number of seconds.
It could indicate which agent has the call, duration, number of mesages, etc.
Agents could ask another (more senior?) agent to assist with a call -- that agent can then review all the messages in the coversation to that point.
An agent could perhaps flag the conversation for further review. Maybe a customer has a bug to report or a new feature to describe, and the conversation should go to the development team for review. Or maybe a customer is being abusive of the agent.

One thing I want to stress is that the conversation should be stored in a database and not be lost to the bit-bucket. Archiving or cleanup can happen at some later date. You know those statements that "this conversation may be recorded for training purposes"? There ya go.