After thawing out frozen connection to unit outside... my unit will not even come on... this house is falling apart, but the thermostat looks fairly new... I have attached pix,

I would check 2 breakers at Electrical Panel, and 2 disconnects are not off or tripped.
If power making it to both units, check 24 volt transformer next.

Inside unit may have a auto type fuse in the circuit board. Check it also.Is anything working? Inside unit? Outside unit? Any noise at all?

Listening is a good idea. Have someone switch thermostat from off to cool, and listen for click of relay in air handler.
Air handler must have 240 volts in, 24 volt transformer must work, and thermostat, and control wiring, for a click. If none, start there.

How do I check for a bad capaciter . How do I check for a bad starter on a/c central air unit

Does anything happen? To get air, the inside blower must come on, the outside compressor must come on, and it won't run long without the outside fan. If nothing happens, likely it is the 24 volt control voltage.

You should have 3 capacitors, maybe a single dual one for the compressor and outside fan. You can do a rough check with an ohmmeter. Short the terminals and remove one wire. Connect the ohmmeter. It should show a high resistance that quickly climbs off scale. Open or short, and replace it.