When using Fly mode (Shift+F in the 3D view), moving the mouse up rotates up, and moving down rotates down. Is it possible to invert this so moving the mouse to the bottom of the screen rotates up?
No, this is not possible out of the box with Blender. In Fly mode, the camera follows the direction of the mouse cursor and rotates accordingly.
Alternatively, you can use R to translate up and F to translate down if the mouse gestures feel odd.
Yes, it is possible. Preferences -> Input -> Walk Navigation -> Reverse Mouse.