There could be several things, or a combination of them, causing the problem. I will need the answers to a few questions, to give you a answer. Can you hear the compressor unit, on the out side of the house, trying to kick on, or making any noise? Is the fan running on the outside unit? Is the cool air coming out or your vents with about the same force the heat does when it is heating your home?
The problem is that power is not getting to the outside unit. This could be caused by a bad connection or other electrical problems. If it were a bad compressor or a problem with the outside unit's compressor, the fan on the outside would still blow. The fan not blowing on the outside unit, but the air blowing on the inside confirms that the thermostat is working. Normally the air blowing out of your vents is at a lesser velocity for air than heat. I am sure the problem is electrical related. Most likely not a homeowner fix, unless they have electrical experience. I am sure if the power is checked on the outside unit it will be found that there is none. Checking for power on the outside unit should be done by a professional. Looks like you have done all the checking you should attempt without electrical training. If you need more help I will be glad to assist you.