Can you barbecue at the river

Yes u can

I've brought a small bbq to the river and used it. Not sure of the actual rules though...

You can in the picnic area but I don't think they allow it on the river area so much because of dry branches and possibly starting a fire.

Yes

There are a few BBQ pits at the picnic/parking area. I'm not sure if you're allowed to bring your own BBQ down to the river though.