Assuming advanced alien technology to create and control wormholes, how big would a wormhole be that would allow a radio signal to pass through? Is there a way to know the size of the mouth as compared to the throat? Also, would such a wormhole mouth put out energy? I've read that wormhole mouths look like black holes when viewed from outside, so would it produce Hawking Radiation?
How big would it have to be? Not very big. Radio waves are electromagnetic radiation, just like visible light. Assuming a focused signal, you'd need an opening about the size of a laser pointer. Maybe a bit bigger to ensure the message gets through.
The rest of your questions.... Well, it's up to you. We have no observational evidence that wormholes actually exist, so you could make them have whatever properties you want.
The wormhole, or any hole for that matter, must be comparable in size to the wavelength of the radiation in order for it to pass through unobstructed. Consider for instance a faraday cage: https://en.wikipedia.org/wiki/Faraday_cage
If you look at the images, the faraday cages depicted are transparent. This is because the distance between the wires of the cage is larger than the wavelength of visible light and so visible light is not blocked. However, longer wavelengths like radio and some regions of microwaves are blocked, which is the desired result.