1. What version of DNN? From 4.3.7 on it (for sure from 4.4.0 on) copies the correct information into the First/Last name fields and had the correct display name. The only thing I can think of off the top of my head is that information isn't in your user's AD profile so the DNN code is putting whatever info it can in there (to prevent errors).
2. What error do you get when you use domain/username. Better yet, try domain\username (note the different "\" mark).
3. Mixed-mode is so that you can have either DNN users or AD users access the site or if you need to make the site available outside your intranet. Under strict Authentication (the Authentication line is uncommented in the web.config) anyone visiting the site on a computer that isn't on your domain will get a pop-up box asking for domain credentials. Mixed-Mode (Authenication line is commented out) allows anonymous users to browse the site or to register and join the site like they could without Authentication.
4. See my answer 1.