My outside ac unit fan will not blow?

All the motors including the compressor on the outside unit have start/run capacitors. These can fail and when they do, the motors wont start

Hummm. A bit difficult to solve this problem. Seems like the condenser outside is not working, I mean the fan of the condenser. Looks like you have a wall A/C and a wall A/C always have condensers outside the house and near the A/C. Have you checked if the fan is spinning? I don't think the compressor is at fault but check it anyway. You should be able to hear the compressor running either in the wall unit or in the outside condenser housing. The only heat you should feel is from the condenser itself. If you blows warm air, that means the fan is working, if not that means the fan is not working. But don't be confused with the fan not working because in winter time sometimes the fan is not working and some times it does. As I said, it's a difficult problem to solve but at least start with checking the situation with the fan in the condenser.

sounds like it is either low on Freon or the compressor is going out. The blowing ice cold is an indicator that it may be freezing up because the low Freon. when it stops blowing cold, turn it to fan only mode and see if it gets colder again. If that is the case then the unit is low on freon and freezing up. I know that sounds confusing but that is how AC works. The low freon pressure means the gas is expanding too fast and the air flow cannot take up the cold from the evaporator fast enough. Turning the compressor off and keeping the fan on will alow the room air to take up the cold.

Check all of the wiring connections from your fan motor. If 1 was OK before, there shouldn't be any reason its not now. Closely check the connections going to the capacitor. If they are back to front, the fan wont start, it will sit there humming. If you give it a flick and the fan starts up, it will be that connection.

Why you need to change evaporator coil when condenser fan not working ,also why would you even charge unit with looking for leak.
Maybe the evaporator coil is leaking and when you recharge it it work for couple of days now unit is empty again .
If unit have low pressure switch then outside unit would shut off.

It is a strong possibility. Use some oil on the motor.It might be the contactor
Before he goes messing with anything inside the unit he needs to kill the power. It may be the capacitor, look for a swollen cap

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The thermostat has a timer on the on- cycle (5 min delay for the outside unit and blower inside) if you are not gonna call a service guy then ( for $25) you can go to homedepot (whatever you got out there) and buy a honeywell vertical digital heat and ac t'stat.
this one doesnt require you to set it up for a particular application. this may solve your problem. but as for warm air out of the registers you need a service guy to check that

All models with ac use freon r134a

Check your coils for ice, check your breaker and fuses for condensing unit, and is your condensing fan running? If not, you either have a bad fan motor or capicitor. After you hit the button, and wait for it to time out, (you should hear a click) take a butter knife, and spin the fan if it is not moving. If the fan start spinning by itself after your assistance, you have a bad capacitor

There are lots of things that could be wrong since he just changed the contactor a wire may have burned...Very common..Make sure all wires feel snug onto terminals. The compressor run capacitor may also be bad it looks like a batter with 2 or 3 terminals if it looks bulged on the top where the wires are than its bad ...if it looks flat you would have to have a meter that checks capacitors to test it. Those are all the easy items to correct..Call for service if all cks ok Please rate my solutuion as solved if you appreciate my expert advise>>>>>>>