Answers

If you do not know what you are doing, hire someone who does!  

Same as others said, Remodeling magazine's Cost Vs. Value report is a great resource!

I'm actually not seeing a wire sticking out of the wall to connect to (in either picture).  There should be a wire sticking out somewhere, unless they also clipped that off, or stuffed it behind the window.  Also, did they leave behind the other half of the sensor?  As others said, wireless may be an option if all else fails in your scenario. 

You definitely want to make sure the person is licensed and that they obtain the proper permits for this.  Most "discount" electric panel change-outs are not done well or properly.  Proper licensing and permitting will protect you.  Try to be home when the inspector comes or contact your municipality directly to verify the inspection was done.  Do this BEFORE paying in full.  

Arc faults can trip easily.  "Updated the electrical" is a broad term.  It is hard to judge if this is substandard work without knowing the scope of work or if permits were obtained.  Did they have permits and inspections?  If the work was permitted and completed poorly, you may have some other options of how to proceed if the electrician remains unresponsive. 

I do not work on pools, but used to have an in ground with vinyl liner in my home.  The first thing that came to mind when I started reading your post was switch to salt water.  Then I read that you mentioned that!  I don't know all the benefits why you should, but I thought the largest were that they are easier to maintain and don't require as much chemical as a chlorinated pool.  Some people I know say that they are much more enjoyable to swim in as well.