Most digital cameras have an AV output. I have a Fuji Finepix s602 and it will allow the AV output to be sending the viewfinder image as it takes the photo, so would be ideal for that application. But as I said, most digital cameras would have that function.
As for the remote trigger, you will probably only find that on the SLR type digital cameras - try the Nikon 10D? or the Canon 30D they retail at around the $1000-$1500NZD range and would do the job well. Also, you can get small flat screen LCD monitors from Jaycar and the likes for around $250NZD.
Hope that helps, PM me if you need more info.