AT&T - TR1909 Trimline Corded Phone - White

Looks like you can from the back. There's a hole to hang on a nail, and looks like you can pop out a piece to make the handset hold onto the front, but it didn't pop out easy to test. I didn't want to break it to see if it would work.

Yes, you can plug a longer cord into the back of the phone. However, it does need to fit through the little opening under the base for the phone to sit flat. It comes with a 2 conductor cord.

Yes, it works with any provider. AT&T is just the make of the phone, not the provider.

I do not see a place to plug in a headset. I have this phone as one of the phones since electricity goes off often where I am and needed a phone not dependent on electricity.

The in-use light does not work without a battery. With battery, red light will flash when another phone on the line is in use.