I had this problem earlier today when someone asked me to chat. The only way I could finally get it to stop was to click "Not Interested" and then before it popped up again, I quickly signed off and logged back in a few minutes later. It took a while and would never just go away if I clicked any of the answers. I had to sign out while the block was gone.. Hope this helps.
There are 2 kinds of chat. One kind of chat is where you engage in private one-on-one conversation with another CM member, sort of like a typed telephone conversation. A private chat may be a good way to have a more spontaneous private conversation before phone numbers are exchanged, but only if you are comfortable doing so. The questions in this thread relate to this type of chat.
For the "Chat Rooms", you can go to the main menu at the top of the page and click "More" on the pull-down menu, then select "Chat Rooms". From there you enter a room, and engage in conversation with others. It is very fast-paced and the topics vary as people ask them. It is sort of like a virtual room with various conversations taking place at the same time.
These are features available to all paying CM members, but you don't have to use them. There other ways to communicate on the site, as you noted.