I'm also confused about which dog is the one with the problem - the new pup or the old dog you've had for a long time.
Always keep the dog under physical control either on a leash or a fenced yard they can't get over or under. With the description of your yard I don't think it's a good idea for them to have the run of the whole thing because of the critters. Critters carry all kinds of nasty diseases and vermin, and you don't want your dogs in physical contact with them. It might be a good idea to invest in a dog run to have in the yard that will keep the critters and the dogs separated.
Some dogs have no big interest in critters while others REALLY REALLY REALLY go batshit crazy over wanting to chase them, catch them and eat them. Seems like both your dogs have the latter obsession. You will probably never be able to curb that deep interest, but you can train them how to behave when they see critters. I envy you the big yard though! What I wouldn't give for a yard like you have! The things you can do with a yard like that! Ooooo! I'm salivating! I'm always looking at other peoples' yards while out walking the dog and I'm just baffled by people with a giant yard that is nothing but a big expanse of grass... no pool, no gardens, no patio, no trees, no nothing. I just don't get it.
Get a prong collar (also called a training collar). Choke chains are dangerous and don't accomplish what they are intended to do (as you have discovered). Here's a lot of good instruction on proper fit and use of a prong collar...
http://leerburg.com/search/searchresults.php?terms=prong%20collarI definitely recommend the Leerburg site (leerburg.com) for all kinds of dog issues. They have a TON of free information right on the site without having to buy any training info they sell. Your dog(s) are not behaving properly because they don't consider you their leader and you likely just don't know how to go about changing that and teaching them to behave appropriately.
You've GOT to stop the escaping and until the dogs are trained not to escape you have to bend over backward ALL THE TIME to prevent it. You can't just throw open a door at this point but need to open it just wide enough to get a toe in and then slide the rest of yourself through so there is no physical way the dog CAN get through it and escape. If it means putting the dogs in another room or into crates just to open the door to go out, so be it. It'll be a pain in the ass until the dogs are trained to not ever escape for any reason, but it's FAR better than them getting hit by a car, causing someone to crash their car or getting into any other dangerous situations for them and for others.
There's some really good videos on the National Geographic site of the Dog Whisperer show that shows how to teach the dog to give up the space in front of the door whether because you want to go through the door or want to answer the door to let someone in. Prowl around the full episodes section for the Dog Whisperer tv program at the site and watch them all. I also remember they had a few shows on the site about dealing with a dog's critter obsession.