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Feel the top of the basket ball sized compressor inside the AC unit sitting outside, if its hot, then that capacitor with the corroded wire is most likely bad.
Let me know if you need coaching on doing that job or not, we can go from there.
Regarding a second filter, the contractor could have chosen to install one, but generally not, one filter is the standard, two restricts the air flow.
The fact that the compressor is not hot AND the contactor is not pulling in, points to a problem in the 24volt control circuit... sometimes a lawn trimmer rodent cuts the wire that attaches the 24v to the outside unit... if that is in tact then the problem is in the electrical or electronic controls.
I can help you trouble shoot that, but it is beyond 'just an answer' and can take hours, tell me you want to try.. In that case I will submit a $30 trouble shooting offer, then after you accept that you can send me photo's of the units various wiring diagrams...
Let me know, we can go from there.