Cats get rid of mice just by existing. We had a mouse problem when we moved in. The cats never once killed a mouse. We no longer have mice (in the house). Mice are always outside so maybe your neighbor just doesn’t like that?

Mice can often find ways you wouldn't believe, but they still have to adhere to the laws of physics, so you might simply not have suitable holes in your apartment for them to enter. But yeah, also quite possible that they can smell the cat and don't want to enter the lion's den.

You didn't HAVE a mouse problem... You may soon, the mice may travel now that their previous source is cut off.

The cat well help though.

*If you pay rent: Tell your landlord, this is their job to fix. Email them about it now so you have proof in case they try to blame you for an infestation...

Yes cat smell can fend off rodents. I only saw rats in my childhood house after my cat died.

You'd definitely find dry little black rice-grain-like droppings on all surfaces if you had mice, especially in rooms that have food, and more will appear if you clean them away. Also, little sticky splotches that are easy to mistake for drink spillages, because they pee everywhere too.

I managed to get rid of mine, but I occasionally still find evidence they were here in out of the way corners that I forgot about.

Brave mice - because I'm sure some will have that trait - and those infected with toxoplasmosis won't care about cats.

Good luck.

Many apartments have a firewall between units, it's a safety measure. If fire can't get through, I doubt mice could.