We are looking at using MQTT as the messaging protocol on a new device we're building. We'd also like a web interface for the device. Does anyone know if you can implement a browser client app (without additional plugins) that talks MQTT?
Yes, as mentioned in Steve-o's comment MQTT via websockets is very possible.
There are 2 options at the moment
To answer your second question lighttpd has a websockets module that can be used to do forwarding to an existing broker with details here.
I've not been able to find anything for Apache that doesn't need you to write your own library to do the forwarding.