Horrendous electric bill in new apartment - what are we doing wrong?
In answer to your first question: Yes. If there is no one occupying an area you should set you T'stat lower while you are out.

In answer to your second question: I have no idea anymore as to what is a fair rate, but I will say your bill makes no sense. We all get the $145.79, but according to the breakdown of your Delivery Services, it only adds up to $110.82 not $133.04. So where does the extra $22.22 come from? Look over your bill and see if it mentions that charge and if it doesn't, I would spend some time on the phone having someone explain it to me. 

On you third question: The first thing I would do is go online, make phone calls, and find any social sites that are discussing this problem. When I say go online and make phone calls, I am talking about directing them to the service provider. I would not try and be rude or mad, but be firm and ask detailed questions about your billing. Also expect and be firm about getting detailed answers.

On matters of saving energy, first as you originally proposed. Set your T'stat to prevent you AC/Heat from coming on as often if you are not in the residence. I use a programmable T'stat and have mine set for 58F in the winter and 80F in the summer. Also I would suggest you don't need it so high if you are sleeping in a bed under covers. I would suggest an electric pad or blanket to keep you warmer in the winter and set the AC for say 60F and come back up to 72 an hour or so before you get up. 

Also orientation is everything as brought up by @1006a. Windows and their exposures are key, you might look and make sure you have heavy insulated curtains to keep conditioned air in especially on extra cold and hot days.

Hope this helps, good luck.