Install skybell doorbell. The old wired doorbell has several sites, about 6 different colors. Not sure which 2 to connect.

If your Honeywell plug-in doorbell won't stop ringing, there are a few things you can try to troubleshoot the issue:
1. Check the batteries: If your doorbell has batteries, make sure they are properly installed and have enough power. Replace them with new ones if needed.
2. Check the ring options: Press the ring option button with the music sign and cycle through the different ring options until you find one that stops the ringing.
3. Reset the doorbell: To reset the doorbell, unplug it from the wall outlet and wait for a few minutes before plugging it back in. This will allow the doorbell to reset and may solve the issue.
4. Check the chime settings: If your doorbell has multiple chime options, make sure the one that's causing the issue is not selected. You can cycle through the chime options using the settings button.
5. Contact customer support: If none of the above steps work, contact Honeywell customer support for further assistance. They may be able to provide additional troubleshooting steps or offer a replacement if the doorbell is defective.

You'll need to connect all the devices to your home's WiFi. But you don't say for sure if the doorbell is wired or not. You said 16 amps, but didn't say what voltage - is that transformer for the doorbell specifically? Once you have downloaded the Nest app to your phone, there should be a help video that can help walk you through the remaining steps. Also, you need to download the Google Home app to your phone to complete the setup.

See if there is sufficient voltage between the two wires using a voltmeter. Also check that there is voltage coming out of the doorbell transformer to the wires, and power from the regular AC line going into the transformer. Sometimes the transformer dies and it's time for a new doorbell, although in the old setups, the transformer was separate from the doorbell.

AC is likely to be the alternating current supply from the transformer.

I suspect two pairs of AC terminals are provided to allow one of two different chimes to be selected with one pair and the common being used for the 4 chime pattern and the other pair and common used for the 8 chime pattern.

The front, side and rear terminals are self explanatory connections from each bell-push with the supply (or return) or other wire from each bell push connected to the COM or common.

The manufacturer's website doesn't feature a wiring diagram though might send you an instruction book on demand. The site does include a note specifying only 18 gauge wire is to be used...

Green should be the Earth, Red should be the live, brown should be black and neutral.

yes it is very simple. All you have to do is connect either the two red wires together or the two black wires together and put the other remaining two on the doorbell chime to the common and front terminals. Here is how it works. One red wire leaves the transformer. It connects to the other red wire going to the pushbutton. When you press the button the voltage comes back to the bell on the black. You have the other black from the transformer on the bell so that completes the circuit. The only reason I say to put either the two blacks or two reds together is that way you're sure to have one wire from the transformer and one from the doorbell. Not a 'pair" of wires from each.

The doorbell buttons and your receiver must be on the same channel. If you will get out the directions that came with your unit, they will describe the process to go through to reset the specific "codes" for your buttons and receiver.

Troubleshooting
Chime does not sound:
• Make sure the doorbell button and the chime codes are the same (See pages
2 and 3).
• Check orientation of doorbell button battery (See page 3).
• Check the doorbell button and chime batteries, replace if necessary.
Batteries seem OK, but the chime does not work when installed:
• Do not mount the chime or doorbell button on metal or near metal studs. This
reduces the transmitter range. Use 1/4" to 1/2" (6 to 13 mm) wood shims to
move the chime or doorbell button off the metal surface.
• Don't mount the chime too close to ground level concrete slabs. This may
also reduce the range. Mount the chime away from the floor.
• Try different locations for the chime closer to the doorbell button. Also try to
orient the chime at a different angle to increase the range.
The range of the doorbell depends on location, temperature and battery
condition.
Regulatory Information
This device complies with Part 15 of the FCC Rules and RSS-210 of Industry
Canada. Operation is subject to the following two conditions: (1) this device may
not cause harmful interference, and (2) this device must accept any interference
received, including interference that may cause undesired operation.
The user is cautioned that changes or modifications not expressly approved by
the party responsible for regulatory compliance could void the user’s authority
to operate the equipment.
Technical Service
Please call 1-800-858-8501 (English speaking only) for assistance
before returning product to store.
If you experience a problem, follow this guide. You may also want to visit our
Web site at: www.hzsupport.com. If the problem persists, call* for assistance
at 1-800-858-8501, 7:30 AM to 4:30 PM CST (M-F). You may also write* to:
HeathCo LLC
P.O. Box 90004, Bowling Green, KY 42102-9004
ATTN: Technical Service
* If contacting Technical Service, please have the following information available:
Model Number, Date of Purchase, and Place of Purchase.
No Service Parts Available for this Product

You will have to have schematics from the old wiring diagram. Then you will have to confirm its consistency with actual old wiring before you can be sure that the wires run to the proper places on the old heater and a/c system. This sounds like a gas furnace and is simpler than a heatpump. Once you know what distinguishes the wires with the same color, you should label them and then you may call them a different color if they were wired with wrong wire colors. You may be in luck if you go to the furnace control box an look at the terminals.. They are usually labeled with a R for Red and W for white and Bl for Blue and B or Black. If you have other colors on these then it sounds that it was wired with wire spare wire by someone who knew what they were doing but neglected to leave a path for someone else follow.
You may be able to use a continuity tester of circuit tester to id these wires that are in question...
ie, once indintified, you can match with the requirements of the new stat.