I have a light set up to a timer that comes on at 6 am and goes off at 6 pm. Before I had this, the egg production dropped for me regardless of all of the light coming in through the visqueen. Around here, it was getting light at 8 and dark at 5:30--definitely not enough daylight for the girls.
As far as the dogs digging under the fence, how much linear fence are you talking? What I did was put that fencing that was real narrow and got gradually larger as it went up (can't remember the name of it) around the outside of my chain link, secured it to the fence and lined it with boulders. Of course my predator proof run (not their daytime fenced in area) is only 12x24 roughly. So it wasn't very expensive to do that. With a larger area, I bet you could go with something cheaper like chicken wire, doubled over, and use that perhaps.
The visqueen is used to provide a wind block. If you only put it 3/4 of the way down, then they'll have wind blowing in no matter where they are in the run. If you don't want to close the entire thing in, then just leave one side (the side less likely to receive wind) open. They need to be able to get out of the wind if necessary and this is a good way to still give them space in the winter and block the wind especially if you're not sure they'll go in at night.