The problem with predators, is that they don't always come around regularly, but they do come around. Also, people that put game cameras up at their coops, often see a lot of animals they didn't even know were coming around at night.
If you search on automatic doors here, there are previous threads with links to companies that sell them. I'm pretty sick today, or I'd look it all up for you. Foy's carries them in the USA. We've ordered and gotten really fast service from Well's in the UK. That order came faster than some of the stuff I've ordered from the US! It's slower ordering directly from the German manufacturer, but we did that once, too.
Ours have worked great for a couple of years. We have the light sensor with probe, as I like to have the door installed on the inside. You can also get a timer, if you'd prefer.
Ok, here's a couple of links:
http://www.foyspigeonsupplies.com/
http://www.foyspigeonsupplies.com/catalog/2100-2107.html
http://www.chicken-house.co.uk/vsb_export.htm
After installation, I just check it to make sure it's working fine, opening and closing at the right times and that the chickens are inside ok at night, the first few days. My chickens didn't have any problem with it, as it moves quite slowly.
I would not leave the coop door open unless your run is made entirely of heavy gage 1/2" hardware cloth, screwed into the wood, including the roof, unless the roof is solid wood and you have either buried wire or used a flange of attached wire all around the perimeter of the run. Anything less and your chickens are at risk.
Those would be my two recommendations, anyway.