Is the transformer bad on my Central Air Conditioner?
I am going to take a stab at this but keep in mind I am a control guy not an HVACR guy.

First let me say a thermostat is nothing more than a set of switches. Your RC is the source side of the power, your C is the return, G and Y turn on your fan and compressor respectively. It doesn't get any easier than that. 

So what we want to do is isolate the problem. First we need to take a voltage reading at the transformer (not the thermostat). In order for any voltage reading mean anything the Common side of the transformer must be grounded. Looking at your previous post I can see a ground lug at your top right. That's where we take our reading. form the xfmr (transformer) Common post (C) to the ground lug. You should be reading zero. Any other reading and you need to ground the C to the ground lug. 

Now that that is done you need to take a reading at the xfmr's secondary side. It should read 24VAC between the C and the RC, and it should read 24VAC from RC to the ground lug. If you don't get 24VAC then you have a bad xfmr.

If you do get 24VAC you then need to move to the T'stat. Take the reading between the RC and C. If do not get 24VAC there you know it's the wiring between the Air Handler and the T'stat.

If you do get 24VAC then you have isolated your problem to the T'stat and beyond. If you take a small jumper and touch the RC to the G you should here the fan come on, and if you then touch it from the RC to the Y you should here the compressor engage. Now set the T'stat so that the AC will come on automatically. Watch and observe, does it work? 

Now you have totally isolated your problem. If both the fan and the compressor come on, the problem is in your T'stat. If either the fan or the compressor doesn't come on then you know you have some other problem down line from the T'stat and that is when I would call a contractor for help.

The problem with your initial readings is that they only show you know how to use a meter. Without the secondary side of the transformer grounded you have no reference, so all you are reading is a floating voltage. I am not saying the system will not work with out a ground. I am saying you can't take any meter readings to analyse you system.

So go through and isolate the problem. Then repair or replace.

Good luck.