This is not true. Assisted GPS doesn't rely on cell networks, it makes use of cell networks for faster fixes. They still work fine without service, but they do take much longer to get a fix. This is evidenced by the fact that you can put your phone in airplane mode and hold it near the window of an airliner and still get a 10-satellite fix.
This is correct. There are a number of signals that GPS receivers use to improve their performance and accuracy. They use both cell-based network location and detection of nearby Wifi access points to get a very fast, rough idea of the location. That enables the system to know what GPS satellites should be in view, which means the GPS receiver doesn't have to wait for as much data from the satellites to get a good location.
They also use Wifi triangulation to fill in gaps in GPS coverage, when they don't have a clear line of sight to the sky. For this reason mobile phones often work much better than dedicated GPS units in cities where the rows of tall buildings reduce visibility of the sky.
They also use the GPS WAAS (Wide Area Augmentation System) when available to help make the GPS location fixes more precise. This system is primarily designed for use by aircraft but it can help ground-based receivers as well.
But you can shut off all of the other stuff, and you phone's GPS will still be able to get a location, as long as it can receive signals from the sky. It'll take longer and may not be as precise, but it will work.