Does this system require me to download an app on my phone, or connect to a local Wi-Fi router to function inside or outside the house?

No. Tiny Traveler Classic does not need Wi-Fi, internet, or a phone app to work. The camera connects directly to the monitor.

Full Answer:
No app, Wi-Fi router, or internet connection is required.
Tiny Traveler Classic uses a direct wireless connection between the camera and the monitor. This means you can use it in the car, at home, in a hotel, or while traveling without connecting it to your phone or local Wi-Fi.
Because the system does not connect to the internet, setup is simple and your video feed stays between the camera and monitor.