How can I wire a single gang 3-way fan control and dimmer?
I don't think it's possible to get everything you want because of the 3-way requirement. There is no true 3-way dimmer setup with double throw switches. The ones promoted as such use electronics to detect the change of state in the satellite switch and mimic 3-way operation. Such a setup may or may not work with your current wiring, you would need to refer to the installation instructions for a specific device.

And even if the pseudo 3-way dimmer is made to work, it cannot be done in a single gang box, the fan control would need it's own box. The closest scheme I can imagine would require a double box at one end with the sole fan control and a separate dimmer control. The other end could be single gang to contain the dimmer satellite control. No fan control would be possible from the satellite end. Even so, you are likely a wire short between fan control and fan, but it's not possible to say for sure without detailed information on all the devices involved.

With most fans available today, it's unlikely to be able to switch the light and fan all with a single pair of 3-way switches. The typical fan needs it's own dedicated speed control. As it's controlled electronically, it's not even possible to set the speed to maximum and then toggle it on and off with a wall switch. When you cut the power, the fan "forgets" it's status. When power is restored, it most likely will default to the off state. And the fan would never work on a dimmer controlled circuit no matter what.

There are other possibilities, but in each case, you will need to give up one or two things, either single gang, dimmer control, use existing 3 wires, or 3-way control. Depending what you would be willing to give up, a solution likely exists. You just can't have it all.