(b) Post links. Everyone in IRC has been doing this for 30 years. Works fine.
(c) Have your client log, or have a bot log to something with HTTP access. You can then use grep or a fancy web app or anything in between to search.
I'm with the parent. Scalable chat is a solved problem; it was solved in the open, via open source and open networks, over 20 years ago. Everything since has been focused not on the technical problems of chat but on the business problem of making money from chat. The fact that someone out there wants to make money on chat does not mean you need to give them money to solve your chat problems. Paying for things of value is fine, but in this case you simply don't need to. Sometimes the best things really are free.