From a hardware POV, what Jabberowocky said. Go hi-fi for the mic/amp sound. As far as the guitar, something that sounds good acoustically is important. Martin has a great video on his site where he goes through all his guitars that's worth watching. He seems to have a different criteria, looking for character in each instrument, and then Martinifying them through the way he plays.

He mics his guitar for live shows, when he can, as well as plugging it in. A large diaphragm condenser mic is his choice. When the sound finally gets to the internet, it's a blend of the guitar and plugged-in sound.

From a personal perspective, I've also found that there are two boxes out there that are great for shaping the electric sound. The Voodoo Labs GIGGITY and Tone Concepts DISTILLERY.

So, in summary, any guitar with a good acoustic sound. An archtop with a recurve, think Benedetto school, will give you a more focused sound, richer in midrange tones. One without a recurve will give a more full range sound, such as those made by Steve Andersen. At least in my experience. As far as down-market guitars, I have found as much variance between two examples of the same model as I have between a flattop and an archtop. Speaking of flat tops, a flat top guitar stands a better chance of giving you the wide frequency range sound you describe.

As far as an amp, go hi-fi.

From there, it's back to the Martin Taylor videos where he tells you exactly how to get that sound using only your hands.