Advice on power section circuit
Center pin should be positive in my experience - experience which includes developing products using that kind of DC power jack. Check Digi-key, Mouser, etc - typical wall-wart suppliers like CUI or Phihong are center-positive by default for barrel connectors. 

Use a pair of diodes to isolate the battery from the plug. The BAT54 type has lower Vf (0.3V) than 1N4001 (0.6V) so loss is less. BAT54 is 1A, but it could be used in place of the BAT46 you have elsewhere in the design. They make 4 different kinds of BAT54: single, common anode, common cathode, and series (BAT54, 54A, 54C and 54S, respectively) so it can reduce component count.

Another choice is center pin size. I used 2.5mm for 5V and 2.1mm for 12V, to prevent plugging a higher voltage to a lower one. 2.1mm would be better for your case.