When I lived with my dad in Biloxi, MIssissippi when I started college, we had a cat that we adopted from vet. We had been out of town and the guy staying at the house to take care of the dog and cat, said that he kept finding lizards and birds in the house. He thought that there must be a hole in the house somewhere. Turns out the cat was climbing the inside of the chimney to get in and out of the house. NEVER HEARD OF THIS BEFORE!!!! So we just added a dog door to the laundry room door in the garage.
I usually woke up with SOMETHING in my bed. Sometimes a lizard (mostly dead - sometimes alive), a bird (always dead), or some kind of toy or item of clothing that struck her fancy. Have you seen the video of the cat that was going out at night and bringing home various pieces of clothing from neighbors' houses? I think she was an early version of that.