extract the flash and eeprom
Have never done that. I have seen solutions posted here about using avrdude commands to read them back out. I've not had luck running avrude commands that way, think I'm missing some simple step.
You cannot read the chip back thru the IDE.
I just keep a copy of the original sketch. Failing that, search the temp folder that the IDE creates on the date you created the sketch, see if you can find the .hex file that was loaded into the promini.
If you select File:Preferencese, and enable verbose outputs, then compile any sketch, you can browse thru the commands that are run and find out where the .hex files are being stored.
You could probably use an Atmel AVR ISP MKii (AVR programmer) and Atmel Studio to read the data back easier via the programming interface (Reset, SCK, MOSI, MISO).