Glad I could help to clear up any concerns you may have. I will happily answer any questions you have based on my own research and experience with HomeKit enabled devices.
Now on to your questions.
1. Since I don't have the Honeywell Lyric thermostat I can't tell you exactly what is involved with that particular device and app but set up is very easy with HomeKit devices. Every HomeKit device has a special code on them. During the installation process you will use the code to add the device to your iOS device. The code can be entered by hand or you can use the iOS device camera to scan the code in for you. HomeKit does not work on Mac OS X computers so a computer is not necessary.
More info here: Use HomeKit-enabled accessories with your iPhone, iPad, and iPod touch - Apple Support
2. There are five (5) apps featured on that page. It would be easier to give the web site address for each manufacturer so you can learn about their HomeKit devices yourself.
NOTE: Some of the apps, like the Lutron Caséta app, will also control non-HomeKit devices. This does not make those devices HomeKit compatible. They will not integrate into the HomeKit ecosystem and may not benefit from all HomeKit has to offer.
iDevices: http://idevicesinc.com/
Lutron Caséta (I currently have this): http://www.casetawireless.com/
ecobee3: https://www.ecobee.com/
Elgato Eve: https://www.elgato.com/en/eve
iHome: https://www.ihomeaudio.com/experience/smarthome/
3. Apple has a constantly updated list of HomeKit compatible devices here: Find accessories that work with Apple HomeKit - Apple Support
Some additional info:
These developer videos will give you an idea of Apple's vision for HomeKit.
Introducing HomeKit
What's New in HomeKit New
You can also get a lot of information by going to https://www.youtube.com/ and searching the name of the devices you are interested in. There are many users posting reviews of HomeKit enabled devices.
Here is one that shows many HomeKit devices used in a home. https://www.youtube.com/watch?v=-2kc6M1fJPE
That video highlights the use of Siri but HomeKit also allows the use of remote access away from the home and automatic triggers based on time-of-day, sunrise/sunset, geofencing to name a few. You can also set up conditional based triggers, for example: If the Smoke/Fire alarm goes off, that can cause all the lights in the home to turn on and the doors to unlock.