Answers

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. 

Mike made a good point.....were any inspections done? What's the age of this home? Size of home? Size of original service v/s new panel? Was the re-wire part of a heavy-up? If done incorrectly, this can lead many problems.

I agree with answer 1.  The electrician should have had his work inspected by an independant electrical inspection agency.  they would place a sticker on the panel bos.  you can also contact the inspection agency yourself and have them do an inspection for you.  It sounds as if the electrician did not follow standard eclectic parctices or code.

You should find someone more competant, the major appliances should be on their own breaker (furnace on a 20a breaker and gas tankless on another 15 or 20a). It can be normal to spread out the circuit recepticals in a few rooms so if a breaker trips in a room there would be another receptical in that same room on another breaker so your not left in the dark. Also the panel should be properly labled. Good luck.