Nighttime Intruder

The first night it happened was about two months ago when Cohen was sleeping between Scott and I. I brought him to bed with us, I wanted to keep an eye on him because he hadn't been feeling well. It was around 2 am and we were all sound asleep, including Izzy who was snoring (yes, snoring) in her bed next to our dresser. I heard the commotion and my heart was pounding out of my chest before I was even fully awake. Izzy had flown from her bed and hit the ground running, barking hysterically, chasing someone or something, out of our bedroom, into the hallway and out of the house to the backyard! I was so startled and extremely frightened, all I could think or say was, "What is going on?" Scott jumped to his feet, more out of it than I was and went to investigate. My body was stiff as I laid in bed next to Cohen, thankful that he was safe next to me. He was so out of it that he didn't even wake up.
Scott didn't find anyone or anything in the backyard and Izzy came back inside and started eating from her bowl in the kitchen just a minute after being outside. We wondered if Izzy was hallucinating or had a night terror that came to life. It took me a while to calm myself down enough to fall back to sleep.
A week or two later, around 9 pm, I sat on the couch while we were watching TV. I heard the doggy door flap back and forth like it does multiple times a day from Izzy going in and out, but then I realized it wasn't Izzy because she was sitting next to me! She jumped off the couch and tore off after whatever had just tried to re-enter our home. Again, we didn't see what it was or find anything that could give us any clues.
This has happened another three times during the night, twice just this week. I have wondered if it was a stray cat, but why would a cat continually enter a house where it gets chased out by a dog? I know that there have been raccoons in neighborhoods close by, but why would it just hang out around here this long?
It is freezing, so whatever this is is probably looking for a warm place to hide out and maybe some food. I love animals and don't want any of them to freeze, but this is getting really frustrating and annoying! I already have one little critter waking me up at night, I don't need another one!
I searched the snow for any other prints other than Izzy's and couldn't find anything that looked different. I noticed that our neighbors behind us has a metal cage trap set in their backyard. I plan on going over and asking them what they are trying to catch, I hope they have some insight. We might just have to close the doggy door at night and let Izzy out when she needs it.
Does anyone have any insight or ideas about what this intruder might be or what we could do to find out?