Creating electricity from heat
As a soldering iron is very hot, thermoelectric generators should easily be able to produce enough power from that. You should be careful, that the generator doesn't get too hot, they have a specified maximum working temperature, so maybe you need a block which dissipates some of the heat.

You will need a power supply circuit, as the voltage of the thermoelectric generator won't fit to your microcontroller.

Energy harvesting power management ICs offer most of the features you want for a project like this, with automated charging of a supercap and outputs which tell you what source is currently powering your circuit.

As concrete product recommendations are rather useless in an answer, I'll just suggest you have a look around at Linear Technologies, they offer some nice solutions for this.