Noel's excellent instructions and your detailed and thoughtful responses should make this pretty easy to solve.
1) I started a New project, In the style picker, I select a style with midi. Say, Clapton1.sty. When I press Quick-play using premade demo, I hear the various midi instruments, but I'm not sure i they are being played back thorugh the Coyote. It sounds like the onboard Microsft synth. Not sure. Anyway...
For your information, you could convince yourself it's the Coyote by clicking the Plugins button on the main screen --> default synth. While the song is playing, you'll see some activity in the Coyote window, e.g. under the Voices column.
2) I load the demo song itself, click ok, return to the main interface and press play. Now all instruments except that style's Realdrums play back as piano. Bass, guitar, and organ parts are all piano. Coyote is showing, but it won't let me change the patch in any slots from Piano.
The only thing I could think of is if you have patch changes disabled in Options | Preferences | MIDI Options. All the "Patch Change" checkboxes on the right should normally be checked.
Fix that problem first (the piano only problem), because it is a bit puzzling, and you want to have a normal working setup before trying to address latency and live playing issues (which may require some experimentation).
3) I can play and hear my keyboard, but only as piano, as per above, and with borderline unuseable latency which I'd never had before.
To improve the latency, some people use ASIO4ALL, and perhaps you used that in the past. To do this click on Audio Settings button, then select ASIO as your audio driver type. If you have ASIO4ALL listed as a device, select it - otherwise you would need to download and install it first and reboot the program. Also, select the "ASIO Always On" checkbox.
4) Loading in the bundled Sampletank, I can load in any instrument in the collection except piano, for exmaples, strings, guitar, you name it, but when I press play, I only hear the same piano I've been hearing. Clicking on Sampletank's virtual keyboard produces no sound. ...
The ASIO solution would probably solve this problem...