Hardlib's answers are 100% on target, but I'll add some thoughts on implementation.
I think a common misunderstanding with MQTT is that it is really a M2M (machine-to-machine) protocol, not a system for exchanging messages between users. That isn't to say you can't use it for messaging (facebook did) but that exists in a layer on top of MQTT. Put another way, MQTT is designed to route messages between machines with little care about the content of those messages. What this means is that user-level niceties (delivery confirmations etc.) aren't really part of it but instead something that you implement on top of MQTT.
So here are some thoughts about how to implement what you propose on top of MQTT:
Consider a situation in which you have two clients (X & Z) which both have access to the same broker (Y). To have client X confirm it has received a message from client Z, simply have client X send a message to a topic (lets say confirmations/z) that client Z is subscribed to. This is trivial to implement in Python or whatever you are writing your application in. (For example, I use that basic procedure to measure round-trip time on my broker.)
However, given that QoS can guarantee that a broker has received the message (and it could be retained or otherwise held for other clients), I would question if this is really necessary unless it is critical that client Z know exactly when client X receives the message.
Depending on your needs there are any number of ways of providing a history for a topic. See the answers here and here for details on MySQL. But if all you need is a local history of a chat or a record of the activity on a few topics, consider simply outputting payloads with timestamps to a text file or JSON. MySQL feels like overkill unless you are dealing with a high volume of messages or need to compose complex queries.