Very sorry for your losses. Poor cat! Poor birds!
Behavior sounds more like coyote. or dog/coyote cross. However, you were there and will be the best judge of what you saw.
You are living near a nature preserve. It is reasonable to assume you will have A LOT of predators in the neighborhood. That will not change even when you remove these particular foxes. You will be killing a lot of predators, possibly illegally (SSS). It will preserve both wildlife and livestock if you protect your domestic animals. If you cannot move away from the nature preserve, it will be your responsibility to protect your livestock from the predators. If you are free-ranging even just during daylight hours, you should probably expect significant losses on a regular basis.
We pasture within an electrified enclosure. This has protected against any ground attacks. We still have to watch for aerial predators.
Yes, guard dogs would also help, if they can be trained to leave your other animals alone.
There is no perfect solution. Everything eats chicken, so it is an on-going problem.
Locations around nature preserves are great fits for people who want to observe a lot of wildlife. Not so great for livestock raising. It is so hard to get people in the places they want to be.
